# NEXUS NETWORK JOURNAL

NEXUS NETWORK JOURNAL

NEXUS NETWORK JOURNAL

### You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong><br />

PATTERNS IN ARCHITECTURE<br />

VOLUME 9, NUMBER 1<br />

Spring 2007

Nexus Network Journal<br />

Vol. 9<br />

No. 1<br />

Pp. 1-158<br />

ISSN 1590-5896<br />

Letter from the Editor<br />

CONTENTS<br />

5 KIM WILLIAMS<br />

Research<br />

7 BUTHAYNA H. EILOUTI and AMER M.I. AL-JOKHADAR. A Generative System for<br />

Mamluk Madrasa Form-Making<br />

31 BUTHAYNA H. EILOUTI and AMER M.I. AL-JOKHADAR. A Computer-Aided Rule-<br />

Based Mamluk Madrasa Plan Generator<br />

59 DIRK HUYLEBROUCK. Curve Fitting in Architecture<br />

71 GIULIO MAGLI. Non-Orthogonal Features in the Planning of Four Ancient<br />

Towns of Central Italy<br />

93 JAMES HARRIS. Integrated Function Systems and Organic Architecture from<br />

Wright to Mondrian<br />

103 CHAROULA STATHPOULOU. Traditional Patterns in Pyrgi of Chios: Mathematics<br />

and Community<br />

Geometer’s Angle<br />

119 RACHEL FLETCHER. Squaring the Circle: Marriage of Heaven and Earth<br />

Conference Report<br />

145 B. LYNN BODNER. Bridges 2006: Mathematical Connections in Art, Music and<br />

Science<br />

Symposium Report<br />

151 SYLVIE DUVERNOY. Guarino Guarini’s Chapel of the Holy Shroud in Turin:<br />

Open Questions, Possible Solutions<br />

Exhibit Review<br />

155 KAY BEA JONES. Zero Gravity. Franco Albini. Costruire le Modernità

LETTER FROM THE EDITOR<br />

One of the most profound relationships between architecture and mathematics is<br />

related to patterns: the patterns of geometric decoration are perhaps the most obvious, but<br />

there are patterns in proportions, patterns in the generation of a series of numbers such as<br />

the Fibonacci series, patterns in the construction of geometric lengths, and patterns in<br />

modular constructions as well.<br />

This issue of the Nexus Network Journal is dedicated to various kinds of patterns in<br />

architecture. Buthayna Eilouti and Amer Al-Jokhadar address patterns in shape grammars<br />

in the ground plans of Mamluk madrasas, religious schools. The Mamluk sultans were a<br />

series of rulers who reigned in Eygpt for nearly 300 years, from 1250 to 1517, and whose<br />

reign saw the creation of very beautiful art and architecture. In their two papers in this<br />

issue, “A Generative System for Mamluk Madrasa Form-Making” and “A Computer-<br />

Augmented Precedent-Based Mamluk Madrasa Plan Generator”, Eilouti and Al-Jokhadar<br />

first compare the significant forms of the ground plans of Mamluk madrasas (by studying<br />

sixteen madrasas built in Egypt, Syria, and Palestine during the Mamluk period) to create a<br />

shape vocabulary, then formulate the operational rules that govern combinations of the<br />

forms (a shape grammar), and finally, create an interactive computerized plan generator.<br />

The shape grammar permits realization of a myriad of patterns based on the initial<br />

vocabulary that all lie within the framework of the madrasa program requirements.<br />

Giulio Magli goes back further in history, to the age of Greek colonies in Italy before<br />

they were conquered by the Romans, to examine patterns in urban design. While most<br />

Greek cities were built in a rectangular pattern, as were the Etruscan and the Roman,<br />

constructed around a set of orthogonal axes (the decumanus and cardus), the settlements<br />

examined by Magli in “Non-Orthogonal Features in the Planning of Four Ancient Towns<br />

in Central Italy” exhibit radial patterns. Magli links this kind of radial urban planning to<br />

the radial patterns of the cosmos.<br />

In “Traditional Patterns in Pyrgi of Chios: Mathematics and Community,” Charoula<br />

Stathopoulou examines the geometric patterns that decorate the buildings of the town of<br />

Pyrgi, on the Greek island of Chios, and uses the research methodology of anthropologists<br />

to examine the relationships between pattern and community there.<br />

“Curve Fitting” is a study of ways to construct a function so that its graph most closely<br />

approximates the pattern given by a set of points. Dirk Huylebrouck’s paper, “Curve<br />

Fitting in Architecture” examines how a pattern of points extracted from an arch, or the<br />

pattern of points that define the curve of a nuclear plant, might be associated to a precise<br />

mathematical curve. Pattern definition in this case could help resolve the issue of whether<br />

or not the architect intended to describe a precise mathematical curve in his or her<br />

construction. For instance, are the arched gates and windows in Gaudi’s Paelle Guell in the<br />

shape of hyperbolic cosines or parabolas? Comparison of the pattern formed by a set of<br />

measured points on the architectural forms with the graphs of the functions could provide<br />

an answer.<br />

Patterns in the architecture of Frank Lloyd Wright have been studied for some time,<br />

both as plan generators, as in the Palmer House, and as decorative motifs, as in his window<br />

designs. Piet Mondrian’s paintings are well-known examples of patterns in art. In<br />

“Integrated Function Systems and Organic Architecture from Wright to Mondrian”, James<br />

Nexus Network Journal 9 (2007) 5-6 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 5<br />

1590-5896/07/010005-2 DOI 10.1007/S00004-006-0026-6<br />

© Kim Williams Books, Turin

Harris looks at the designs of these two masters to extract the rules of their pattern<br />

generation and propose possible applications.<br />

In the Geometer’s Angle column, Rachel Fletcher examines geometric constructions that<br />

square the circle. Her “Squaring the Circle: Marriage of Heaven and Earth” looks at the<br />

combination of square and circular patterns.<br />

This issue is completed by B. Lynn Bodner’s report of the annual Bridges conference,<br />

the 2006 edition of which took place in London in August of this year, Sylvie Duvernoy’s<br />

report on the symposium “Guarino Guarini’s Chapel of the Holy Shroud in Turin: Open<br />

Questions, Possible Solutions”, and by Kay Bea Jones’s review of the exhibit “Zero Gravity.<br />

Franco Albini. Costruire le Modernità” in Milan.<br />

I think that this is one of the best issues ever of the Nexus Network Journal and I hope<br />

you enjoy it.<br />

6 KIM WILLIAMS – Letter from the Editor

Buthayna H. Eilouti<br />

Department of Architectural<br />

Engineering<br />

Jordan University of Science and<br />

Technology<br />

POB 3030<br />

Irbid 22110, JORDAN<br />

buthayna@umich.edu<br />

Amer M. Al-Jokhadar<br />

College of Architecture & Design<br />

German-Jordanian University<br />

Amman, JORDAN<br />

TURATH: Heritage Conservation<br />

Management and Environmental Design<br />

Consultants<br />

P.O.Box: 402, Amman, 11118 JORDAN<br />

amerjokh@hotmail.com<br />

amerjokh@gmail.com<br />

Keywords. Shape grammar, generative<br />

system, visual studies, Mamluk<br />

architecture, school design, Islamic<br />

architecture, design process<br />

Research<br />

A Generative System for Mamluk<br />

Madrasa Form-Making<br />

Abstract. In this paper, a parametric shape grammar for<br />

the derivation of the floor plans of educational<br />

buildings (madrasas) in Mamluk architecture is<br />

presented. The grammar is constructed using a corpus<br />

of sixteen Mamluk madrasas that were built in Egypt,<br />

Syria, and Palestine during the Mamluk period. Based<br />

on an epistemological premise of structuralism, the<br />

morphology of Mamluk madrasas is analyzed to deduce<br />

commonalities of the formal and compositional aspects<br />

among them. The set of underlying common lexical<br />

and syntactic elements that are shared by the study<br />

cases is listed. The shape rule schemata to derive<br />

Mamluk madrasa floor plans are formulated. The sets<br />

of lexical elements and syntactic rules are systematized<br />

to form a linguistic framework. The theoretical<br />

framework for the formal language of Mamluk<br />

architecture is structured to establish a basis for a<br />

computerized model for the automatic derivation of<br />

Mamluk madrasa floor plans.<br />

1 Introduction<br />

On the one hand, the study of design, its underlying representations, and the methods<br />

that can be used to derive new artifacts are important research topics in many disciplines,<br />

including engineering and architecture. Form-making entails design activities that have a<br />

direct influence on the appearance of the artifacts produced. Its study involves the<br />

establishment of explicit and systematic links between the form of an artifact, its visual<br />

properties, its compositional attributes and its generative considerations. In addition, formmaking<br />

is concerned with the processes and considerations that precede and follow as well<br />

as those that produce the final form. Shape grammar represents a systematic method for<br />

studying the form-making layer of design activities. Studies in the area of shape grammars<br />

are well established. In many applications they proved to be powerful in shape derivation,<br />

analysis and prediction. However, many of their potentials are still far from being fully<br />

explored, especially in the area of understanding the morphology of architectural<br />

precedents.<br />

On the other hand, Mamluk architecture represents a significant period of Islamic<br />

architecture. It displays most of the aesthetic principles that underlie Islamic architecture.<br />

Most of the aesthetic values of Mamluk architecture are exhibited in Mamluk educational<br />

buildings (madrasas). The morphological structure that underlies the forms of Mamluk<br />

madrasas can be mathematically analyzed and syntactically systematized to formulate a<br />

powerful compositional language that may help in the understanding of the Mamluki style<br />

and the aesthetic principles of Islamic architecture.<br />

Nexus Network Journal 9 (2007) 7-30 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 7<br />

1590-5896/07/010007-25 DOI 10.1007/S00004-006-0027-5<br />

© Kim Williams Books, Turin

There is no systematic study of the formal aspects of Mamluk architecture that explicitly<br />

articulates the compositional language that underlies its design, and the procedural<br />

sequence of its form generation.<br />

In this paper, the two strands of shape grammar implementation and Mamluk madrasa<br />

morphological investigation are studied and interlaced. The connection between the two<br />

strands produces a new language for the formal composition of Mamluk madrasas. This<br />

language summarizes a morphological analysis of Mamluk educational buildings in a<br />

concise generative system that can be used to derive emergent examples of this significant<br />

period in the history of Islamic architecture.<br />

2 Background<br />

2.1 Formal Languages<br />

Shape grammar is that part of design study which deals with the morphology of the<br />

overall forms of products and their internal structures and with the incremental processes<br />

that generate them. It is concerned with the constituent components of a form and their<br />

arrangements and relationships. It usually emphasizes the lexical level (vocabulary elements)<br />

and the syntactical level (grammars and relationships) of the architectural composition,<br />

rather than the semantic, symbolic or the semiotic levels. When rules of a shape grammar<br />

are executed on a given subset of vocabulary elements, different alternatives of artifact<br />

forms are created. The set of all forms that are generated by applying given rules on given<br />

vocabulary elements constitutes a formal language, which may correspond to a visual style.<br />

Studies in the area of shape grammar started in the early 1970s, but their roots in<br />

pattern language, typology and systematic design methods started earlier. Although shape<br />

grammars have good predictive, generative, derivative, and descriptive powers, they do not<br />

focus on the historical, social, functional, or symbolic aspects of the architectural<br />

compositions.<br />

Using a step-by-step process to generate a language of design, a shape grammarian<br />

typically formulates a set of shape rules, that is, transformations and parameters that can be<br />

applied to a set of given vocabulary shapes in order to reproduce existing shapes and come<br />

up with emergent ones. Different types of shape grammars can be defined according to the<br />

restrictions that are used in their applications. They vary according to the format of rules,<br />

variable parameters, constant proportions, allowed augmented attributes, and the order of<br />

rule applications. All components of shape grammars (vocabularies, spatial relationships,<br />

parameters, attributes, rules, transformations, and initial shapes) provide a foundation for a<br />

science of form-making and for a theory of systematic architectural design and composition<br />

methodology through algorithms that perform arithmetic calculations on geometric shapes.<br />

Stiny [1980] proposed a framework to define a language of design, constructed by<br />

means of shape grammars. The framework can be developed through applying the<br />

following five stages [Osman 1998]:<br />

1. Vocabulary definition. The basic shapes of a formal language are defined. They<br />

function as the basic building blocks for design.<br />

2. Spatial relationship determination. In this stage, the structure observed in a set<br />

of designs is investigated to deduce the spatial relationships that are used to<br />

connect the building blocks.<br />

8 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

3. Rule formulation. Shape rules are formulated in terms of the spatial relationships<br />

identified in the second stage on the basic shapes listed in the first stage.<br />

4. Shape combination. Shapes in the vocabulary set are combined to form initial<br />

and subsequent shapes. Shape rules are applied recursively to initial shapes to<br />

generate new shapes.<br />

5. Shape grammar articulation. Grammars are specified in terms of shape rules,<br />

initial shapes and new shapes. Each shape grammar defines a language of design.<br />

2.2 Mamluk madrasas<br />

Visually speaking, Mamluk architecture represents a significant period of the Islamic<br />

architecture. It exhibits the major goal of Islamic architecture, according to which designers<br />

strive to achieve harmony between people, their environment and their creator. In general,<br />

there are no strict rules defined to govern Islamic architectural design. Designers of the<br />

major institutional buildings of Islamic cities used local materials and construction methods<br />

and applied abstract geometric languages to express in their own ways the character, order,<br />

integrity, harmony, and unity of architecture. However, when the great monuments and<br />

precedents of Islamic architecture are examined, their formal structure reveals a complex<br />

system of geometrical relationships, a well-designed hierarchy of space organizations and a<br />

highly sophisticated articulation of ornaments, as well as deep symbolic and semantic<br />

connotations. Geometry in Islamic architecture was developed into a sacred science. It has<br />

been structured to express the Islamic beliefs and views of the relationships between the<br />

world, man, and God [Himmo 1995]. In the Islamic perspective, the method of deriving<br />

all the organizational proportions of a building form from the harmonious recursive<br />

division of a basic shape is a symbolic way of expressing the oneness of God and his<br />

presence everywhere [Himmo 1995]. Compositions in the Islamic architecture have been<br />

transformed into highly abstracted shapes on which principles of rhythmic repetitions,<br />

unity, symmetry, and variation in scale were applied to create ordered yet dynamic effects.<br />

Shape in Islamic architecture is strongly related to the study of mathematics and other<br />

sciences.<br />

In Islamic culture, education has always been closely connected to worship. Expressing<br />

this, from their beginnings Islamic mosques have been used for both praying and learning.<br />

Over time, the mosque experienced various transformations of functions, which resulted in<br />

the establishment of a number of related building types of social, educational and religious<br />

characters and with narrower functional scopes. The two interrelated functions (worship<br />

and teaching) of mosques eventually diverged. The separation resulted in a distinguished<br />

sacred mosque and a madrasa. The early madrasa buildings offered special open and closed<br />

teaching halls. The form and function of the early madrasas were similar to those of the<br />

mosques. Eventually, the architectural typology of the madrasa forms became so prominent<br />

that it visually influenced mosque architecture in the Islamic region during the twelfth to<br />

the fifteenth centuries A.D. [Bianca 2000]. The two main prototypes of madrasa layout<br />

shapes are the open courtyard madrasa and the closed or domed courtyard madrasa (fig. 1).<br />

The domed madrasas are usually smaller buildings whilst those with an open courtyard are<br />

generally larger and have central Iwans surrounded by arcades.<br />

The most common formal prototype of the early madrasas is the four-Iwan plan. An<br />

example of this prototype can be seen in the Mustansriyya Madrasa in Baghdad. Although<br />

it is traditionally thought that madrasas provided sleeping and working accommodation for<br />

students, the extant examples show that this was not a rule and it is only later on the<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 9

progress line of madrasa buildings that student facilities became accepted parts of a madrasa<br />

design [Bianca 2000].<br />

Fig. 1. The main madrasa layout types: a, above) The open courtyard<br />

madrasa type (al-Sultan Qalawun Madrasa in Cairo); b, below) The roofed<br />

or domed courtyard madrasa dorqa’a type (al-Sultan Inal Madrasa in Cairo).<br />

Reproduced by authors, from [Organization of Islamic Capitals and Cities<br />

1990]<br />

3 The morphology of Mamluk madrasa designs<br />

3.1 Case studies of Mamluk madrasas<br />

In order to develop a shape grammar for Mamluk madrasa design, a sample of sixteen<br />

cases has been assembled (Table 1). The sample consists of thirteen madrasas in Cairo, two<br />

10 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

madrasas in Jerusalem, and one madrasa in Aleppo. The madrasas in the sample have been<br />

selected for their important role as representatives of Mamluk architecture.<br />

Table 1: The study sample for Mamluk madrasas classified according to their geographic locations<br />

3.2 Common compositional features of Mamluk madrasas<br />

Most of the Mamluk madrasas in the sample were erected on restricted sites. The<br />

exterior layouts of these madrasas respected the shape of the site they were constructed on.<br />

Thus, irregular ground floor plan shapes were almost always generated. However,<br />

considerable thought and effort were often given by the designer to make the building<br />

regular in shape inside. Basic shapes were used as the basis for generating all interior spaces.<br />

Most of the major interior spaces were oriented toward the “Qibla direction” (the prayer or<br />

Mecca direction). Intermediate spaces appeared between the perfectly regular shapes of the<br />

interior spaces and the irregular outer boundary of the site.<br />

Typically, the aforementioned four-Iwan madrasa was a dominant prototype. In this<br />

prototype, the four Iwans (South-Eastern Iwan or “Qibla Iwan”, North-Western Iwan,<br />

South-Western Iwan, and North-Eastern Iwan) surround the central courtyard. The other<br />

spaces were located on the sides. In addition to each Iwan, facilities for many functions<br />

have been designed: a residential unit for the sheikh (teacher), small units for students,<br />

small court, sabil (free water fountain), minaret (tower), the tomb for the patron of the<br />

madrasa, corridors and transitional spaces, sadla (secondary Iwan), ablution space, and<br />

water closets.<br />

Analyzing the sixteen case studies reveals the following major distinct elements:<br />

– The dominant portal space.<br />

– Great Iwans with vaulted roofing.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 11

– Huge interior courtyards or Sahns.<br />

– Sheikh’s and students’ cells.<br />

– Large spaces covered with stone vaulting.<br />

– Minarets that emphasize the portals.<br />

3.3 Mamluk madrasa typology<br />

In general, the floor plan of Mamluk madrasa exhibits a logical design with a vigorous<br />

articulation of clearly distinct elements. The design components include great teaching<br />

Iwans with tierce-point stone vaulting, huge interior courtyards, mausoleums under high<br />

cupolas, residential cells and polygonal minarets [Stierlin 1984]. The overall plan is shaped<br />

as a rectangular courtyard with an Iwan in the center of each side. Teaching takes place in<br />

the Iwans, and the students live in the cells arranged along the intermediate walls. The<br />

dominant architectural feature of this typology is the four Iwans built into the center of<br />

each courtyard side.<br />

Later Mamluk madrasas tend to rise vertically in a number of stories, as opposed to the<br />

flat horizontal expansion of the early ones, which provided space for the multitudes on the<br />

same ground level. In addition, they tend to attach subsidiary units such as tomb chambers<br />

and sabil-kuttab to the main madrasa functions, converting their simple forms into large<br />

funerary complexes [Parker 1985].<br />

3.4 Bahri and Burgi Mamluk madrasas<br />

Mamluk sovereignty can be broadly classified into two periods: the Bahri, which ruled<br />

from 1250 A.D. to 1382 A.D., and the Burgi, which ruled from 1382 A.D. to 1512 A.D.<br />

[Parker 1985]. Mamluks spread extensively in Cairo, but they also reached Jerusalem and<br />

Aleppo.<br />

Madrasas in the Bahri Mamluk Period are characterized by the following features (fig.<br />

2a):<br />

– The four Iwans surround the open courtyard, while other spaces are located on<br />

the sides and mainly in the first floor. This type has been used for teaching one<br />

or more legal rites;<br />

– In addition to each Iwan, many facilities have been designed. These include a<br />

residential unit for the Sheikh (teacher), small units for students and small<br />

courts;<br />

– In most madrasas, there was a tomb for the patron of the madrasa and his<br />

family;<br />

– The portal space has been marked and emphasized by a minaret above it;<br />

– The ablution and water closets are located in the back of madrasa for<br />

ventilation, and oriented to face the sun. Their level is lower than that of the<br />

madrasa itself.<br />

Madrasas in the Burgi Mamluk period are characterized by the following features (fig.<br />

2b):<br />

– The madrasa consists of large and central open courtyard surrounded by four<br />

Iwans. These Iwans were divided into riwaqs (roofed halls), or covered by<br />

either pointed vaults or wooden roofs;<br />

12 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

– The ground floor plan includes spaces for the family of the patron of the<br />

madrasa, and rooms for teachers and students;<br />

– Another type in this period is the roofed courtyard dorqa’a which has been<br />

covered by a Shokhsheikha (wind ventilator) instead of the open court;<br />

– The two lateral Iwans have been replaced by smaller Iwans;<br />

– Most madrasas have a “sabil” (free water fountain);<br />

– In many examples, the whole madrasa was used to teach one legal rite;<br />

– The portal and vestibule space have been also marked and emphasized by a<br />

minaret above the main gateway;<br />

– The ablution and water closets have been located in the back of madrasa.<br />

Fig. 2. Layouts of madrasa floor plans: a) in the Bahri Mamluk Period, b) in<br />

the Burgi Mamluk Period. Reproduced by authors, from [Organization of<br />

Islamic Capitals and Cities 1990]<br />

Mamluk architects repeatedly used certain features in their buildings that shared some<br />

common attributes in size and type. Thus, close relationships can always be found among<br />

these buildings in their proportions, geometric shapes, topological relationships of spaces<br />

and formal compositional aspects. Some of the common features that can be observed in<br />

the Mamluk madrasas are:<br />

– The geometric features and proportions of the great and small arches of<br />

building’s Sahns (central courts of madrasa) are similar;<br />

– The position of Iwans may be identified by the fenestrations of the façades of<br />

the madrasa without actually entering the building;<br />

– Externally, fenestrations, whether alone or in groups, were set back from the<br />

wall planes in recesses. The dimensions and shapes of the openings and recesses<br />

reflect the position and size of the hidden Iwan and provide the details<br />

necessary to identify the Qibla (Mecca orientation) Iwan;<br />

– The massing configurations illustrate apparent recourse to asymmetrical<br />

compositions.<br />

– The façade treatment exhibits skillful articulation of openings and detailing of<br />

ornaments;<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 13

– The case study sample displays well-defined vocabulary components such as<br />

gateways, courtyards, Iwans, domed funerary hall and minarets;<br />

– Locations of the main components such as court(s) and Iwan(s) are similar;<br />

– A proportional system of individual components and their combinations<br />

underlies the case studies;<br />

– The overall organizational rules that govern combining the spaces are based on<br />

the principles of symmetry, axis position and rotation, and the topological<br />

relationships of spaces.<br />

4 A generative system for the Mamluk madrasa floor plans<br />

A morphological analysis of Mamluk madrasa architectural examples reveals the<br />

existence of underlying coherent geometric language on all scales. Within this language,<br />

architectural vocabulary elements along with spatial organization rules and aesthetic<br />

principles of the components and their compositions are defined.<br />

The resultant formal language can be represented as a parametric shape grammar that is<br />

based on a set of points, lines, and labels. Multiple aspects of the formal language can be<br />

parameterized. The parametric attributes include the size and angle of the rooms, the<br />

position at which courtyard walls are attached to rooms, the size of the courtyard and the<br />

arrangement and size of the elements of the building. Such a parameterization of rules<br />

significantly decreases the number of the proposed rules. It also increases the predictive and<br />

derivative powers of the formulated grammar.<br />

4.1 The vocabulary elements of Mamluk madrasas<br />

In making any architectural statement, the designer calls upon a formal vocabulary<br />

drawn from his or her previous experience and from the background tradition or culture in<br />

which the design is being executed [Serageldin 1988]. Among the institutions (khanqah,<br />

ribat, and madrasa) that were common in Mamluk architecture the one that describes most<br />

of its architectural vocabulary is the madrasa. The spatial vocabulary elements shared by the<br />

case studies are illustrated in fig. 3.<br />

4.2 The Grammar of Mamluk madrasas<br />

4.2.1 The procedure of grammatical rule formulation. The grammar rules are formulated<br />

according to the following considerations:<br />

– Rules are formulated to specify how sub-shapes of a composition in progress<br />

will be replaced by other shapes.<br />

– A rule applies if there is a similarity transformation that will bring the shape on<br />

the left-hand side of a rule into coincidence with a sub-shape in the shape<br />

vocabulary list.<br />

– Labels are alphabetic characters that are associated with points and shapes.<br />

These labels are used to control the application of rules. An example of these<br />

labels is the symbol associated with the qibla wall which controls the Qibla<br />

direction of madrasa spaces.<br />

– Shapes have proportional parameters which will be assigned by the grammar<br />

description during the process of its application.<br />

14 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

Fig. 3. The Vocabulary Elements of the Plan of al-Thaher Barquq madrasa,<br />

Cairo: 1– Courtyard (or dorqa’a); 2–North-Western Iwan (opposite to Qibla<br />

Iwan); 3–Derka “entrance spaces”; 4–Small Cells for the Students; 5–Main<br />

Entrance and portal space; 6–Corridors and Transitional Spaces; 7–Ablution<br />

Space; 8–Sabil (free water fountain); 9–South-Eastern Iwan (Qibla Iwan);<br />

10–Southwestern and Northeastern Iwans; 11–Tomb or Mausoleum for the<br />

madrasa’s patron; 12–Minaret (see fig. 4); 13–Teacher’s House (Sheikh’s<br />

House); 14–Northern and Southern secondary Iwans (Sadla). (Reproduced<br />

by authors, from [Organization of Islamic Capitals and Cities 1990]<br />

Fig. 4. A minaret that is: a) aligned with the street, while the rest of the building follows the angle<br />

of Mecca orientation, b) aligned with Mecca orientation<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 15

The morphological aspects, which include visual principles such as symmetry,<br />

proportion, geometry, axes distribution, addition and subtraction of spatial organizations,<br />

and shape transformations, are considered in the shape grammar development. These<br />

aspects are illustrated in fig. 5.<br />

In general, the grammar derives the plans starting from organizing the exterior layout<br />

shape and the interior spaces, and proceeding down to the details of walls, doors, and<br />

windows. The grammar is formulated in a way that encourages a designer to start with the<br />

determination of the exterior layout. This point of departure is more efficient for<br />

controlling the overall shape, because the interior layout is more constant in its geometric<br />

shape and orientation. The overall structure of the formulated grammar is illustrated in fig.<br />

6. The computational model is out of the scope of this paper. Its study is presented in a<br />

separate paper [Eilouti and Al-Jakhadar 2007].<br />

Fig. 5. Aspects used for developing a generative system for Mamluk madrasas<br />

16 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

Fig. 6. A flowchart illustrating the different stages for developing a generative system for Mamluk<br />

madrasa floor plans<br />

4.2.2. The grammar rules for generating Mamluk madrasa floor plans. All rules of Mamluk<br />

madrasas are numbered using the system X – n – Y , where:<br />

X: is the name of the stage. The possible values of this variable are:<br />

SL (Schematic Layout)<br />

AC (Architectural Components)<br />

WT (Walls Thicknesses)<br />

OP (Openings)<br />

TR (Termination)<br />

n: is the rule’s number in stage X.<br />

Y: describes the sequential step that is associated with the rule (n). The possible values of<br />

this variable are:<br />

PSH4 (the addition of a Parametric Shape with Four Sides)<br />

or PSH5 (the addition of a Parametric Shape with Five Sides)<br />

or PSH6 (the addition of a Parametric Shape with Six Sides)<br />

or PSH7 (the addition of a Parametric Shape with Seven Sides)<br />

or PSH8 (the addition of a Parametric Shape with Eight Sides)<br />

or PSRF (the addition of a Parametric Shape and then Reflect it)<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 17

or A (the addition of a Parametric Shape - Arc)<br />

or C (the addition of a Parametric Shape - Circle)<br />

or M (Moving)<br />

or R (Rotation)<br />

or D (Division)<br />

or S (Scaling)<br />

or U (Union)<br />

or L (Drawing Lines)<br />

There are three steps for Mamluk madrasa floor plan form generation. These are:<br />

1. Initial Shape Location:<br />

The initial shape, from which all plans are generated, establishes a point labeled “A” at the<br />

origin of the coordinate system, as shown in fig. 7.<br />

Fig. 7. The initial shape from which all Mamluk madrasa floor plans are generated: a labeled<br />

axis.<br />

2. Exterior Layout Shape Determination:<br />

The exterior layout shape is generated around the initial shape. Six alternatives for this<br />

set are defined. These are illustrated in fig. 8.<br />

3. Grammar Rule Application:<br />

Fig. 8. The Exterior Layout Shapes<br />

The formulated grammar consists of 93 rules distributed as follows:<br />

Rule type<br />

Number of Rules<br />

Rules for generating schematic external layout shapes 12<br />

Rules for generating architectural components:<br />

Rules for generating interior spaces 2<br />

Rules for determining the orientation of spaces 1<br />

Rules for generating Iwans and courtyards 3<br />

Rules for scaling Iwans and courtyards 3<br />

Rules for generating spaces between Iwans 2<br />

Rules for generating lateral spaces 4<br />

Rules for generating mihrab in the Qibla-Iwan 3<br />

Rules for articulating interior spaces 4<br />

18 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

Rules for generating the tomb or mausoleum 13<br />

Rules for generating ablution spaces 2<br />

Rules for generating cells and rooms around the courtyard 4<br />

Rules for generating the main entrance and Derka 18<br />

Rules for determining wall thickness 9<br />

Rules for generating openings 9<br />

Termination rules 4<br />

5 The derivation process of Mamluk madrasa floor plan<br />

In order to demonstrate the shape grammars developed so far, an example is illustrated.<br />

It is selected from Type 2 (see fig. 1b), which has a roofed courtyard dorqa’a. This case is<br />

al-Ashraf Barsbay Madrasa in Cairo (1425 A.D.). The main components of this building<br />

are the central open courtyard surrounded by two large Iwans. The Qibla-Iwan is the<br />

largest one. There are also two other small Iwans, sadlas. The total area of this madrasa is<br />

approximately 1550 m 2 , and the area of the central courtyard is about 230 m 2 . Thus, the<br />

proportion between the total area and the area of the courtyard is 1:6.7. Other spaces that<br />

are represented in this madrasa are the mausoleum in the Eastern-North side, sabil-kuttab,<br />

and a minaret above the main portal.<br />

The main entrance of this example is characterized by the vestibule space. The main<br />

gateway of the madrasa does not allow immediate access to the indoor spaces, but leads into<br />

a passage with a 90° turn, so that it is impossible to see the courtyard from the outside.<br />

This indirect access enhances the creation of calm learning environment inside. Another<br />

compositional feature is the division between the ablution space and other rooms. The<br />

main reason for this is the orientation to the wind, ventilation and sun.<br />

The derivation process of this example is organized into five major stages. These consist<br />

of the generation of the exterior layout, the generation of architectural components, the<br />

determination of walls thicknesses, the assignment of the openings – doors and windows–,<br />

and, finally, the rule termination stage. Within these stages ten more detailed stages can be<br />

identified. Figs. 9-12 illustrate the ten stages through which the overall ground floor plan of<br />

al-Ashraf Barsbay Madrasa was derived. The ten stages are:<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 19

1. Determining the shape of the Exterior Layout by applying the rules: SL-1-<br />

PSH4, SL-2-A and SL-13-M (fig. 9a):<br />

Fig. 9a<br />

2. Outlining the Overall Interior Spaces by applying the rules: AC-3-R, AC-2-M,<br />

AC-1-PSH4 (fig. 9b):<br />

Fig. 9b<br />

20 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

3. Generating the overall interior spaces by applying the rules from (AC-4-D) to<br />

(AC-9-S) (fig. 9c):<br />

Fig. 9c<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 21

4. Generating the courtyards, Iwans, and minbar by applying the rules from AC-<br />

10-PSRF to AC-21-PSH4 (fig. 10a):<br />

Fig. 10a<br />

22 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

5. Generating mihrab, sadla, and tomb by applying the rules from AC-16-A to<br />

AC-24/b-C (fig. 10b):<br />

Fig. 10b<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 23

6. Generating ablution space, and cells around the courtyard by applying the rules<br />

from AC-27-PSH4 to AC-32-U (fig. 11a):<br />

Fig. 11a<br />

24 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

7. Generating the entrance and sheikh’s house by applying the rules from AC-<br />

33/a-PSH4 to AC-40-U (fig. 11b):<br />

Fig. 11b<br />

8. Generating the overall layout of al-Ashraf Barsbay Madrasa after applying the<br />

rules for determining the thicknesses of walls by applying the rules from (WT-<br />

2, WT-5, WT-6, WT-7, WT-8, and WT-9) (fig. 12a):<br />

Fig. 12a<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 25

9. Assigning openings for Qibla Iwan by applying the rules OP-1, OP-2, and OP-<br />

7 (fig. 12b):<br />

Fig. 12b<br />

10. Assigning openings for tomb by applying the rules OP-8 and OP-9 (fig. 12c):<br />

Fig. 12c<br />

As a result of the ten-stage derivation process the floor plan of al-Ashraf Barsbay madrasa<br />

is produced. fig. 13 illustrates the floor plan after the rule applications in the ten stages:<br />

Fig. 13<br />

26 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

6 Conclusion<br />

In this paper, a new framework that systematically describes the morphological structure<br />

and the derivation process of Mamluk madrasas is introduced. The framework represents<br />

an explicit and externalized process for generating the forms of architectural precedents.<br />

Within this framework, a formal language for the Mamluk madrasa floor plans is<br />

formulated. A set of vocabulary elements is concluded from a deductive morphological<br />

analysis of a group of study cases. The study sample includes sixteen madrasas selected from<br />

Egypt, Syria, and Palestine. These Mamluk madrasas share common features and<br />

vocabularies in plan composition. The features include location of components such as<br />

courtyard, Iwans, tomb, sabil and minarets; exterior layout shape, orientation and<br />

topological relationships. Visual aspects such as proportion, symmetry, axial organization,<br />

and rotation are also shared by the study sample instances.<br />

A set of grammatical rules is formulated to guide the sequential process of composing<br />

the vocabulary components into a meaningful plan that can be considered a member of the<br />

family of Mamluk architectural styles. The grammar set consists of ninety-three rules for a<br />

step-by-step derivation of a plan of Mamluk madrasas. The process of generating a floor<br />

plan for a Mamluk madrasa involves applying rules in many phases. These include rules for<br />

generating schematic external layout shapes, architectural components, interior spaces,<br />

Iwans and courtyards, spaces between Iwans, lateral spaces, mihrab in the Qibla-Iwan,<br />

Tomb or Mausoleum, ablution spaces, cells and rooms around the courtyard, the main<br />

entrance, derka and openings. In addition, there are rules for: determining the orientation<br />

of spaces, scaling Iwans and courtyards, articulating interior spaces, determining wall<br />

thickness and, finally, the termination rules.<br />

In order to develop the formulated formal language, this research adopted two main<br />

methods: the case study approach (which focused on the educational buildings in Mamluk<br />

architecture); and the theoretical research method (represented by the deductive analysis<br />

and the morphological study, as well as the mathematical and the linguistic representations<br />

for developing shape grammars).<br />

The theoretical framework is mathematically and geometrically developed to explore<br />

new layers in the design of beautiful Mamluk buildings. Possible future extensions may<br />

focus on a morphological study and design of a generative system for the facades, the threedimensional<br />

massing organizations, the decoration articulation or the opening treatment of<br />

Mamluk buildings. Further, our research has been concerned with the development of the<br />

formulated framework into computer-aided precedent-based educational software that<br />

trains students to discover new aspects and layers of existing Mamluk architectural<br />

examples and explore emergent designs that can be considered new imaginary additions to<br />

the same architectural style applied possibly to recent functions [Eilouti and Al-Jokhadar<br />

2007].<br />

References<br />

AL-JOKHADAR, A. 2004. Shape Grammars: An Analytical Study of Architectural Composition Using<br />

Algorithms and Computer Formalism, Unpublished Master Thesis, Jordan University of Science<br />

and Technology, Jordan.<br />

BIANCA, S. 2000. Urban Form in the Arab World: Past and Present. London: Thames & Hudson<br />

Ltd.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 27

BURGOYNE M.H. 1987. Mamluk Jerusalem: An Architectural Study. Published on behalf of the<br />

British school of archeology in Jerusalem by the Islam Festival Trust, London.<br />

CHOMSKY, N. 1957. Syntactic Structures. The Hague: Mouton and Co.<br />

DING L., GERO J.S. 2001. The emergence of the representation of style in design. Environment and<br />

Planning ‘B’: Planning and Design 28: 707- 731.<br />

EILOUTI, B. 2003. Three-dimensional modeling by replacement. In: Proceedings of the IASTED<br />

International Conference: Modeling, Simulation, and Optimization (July), Banff, Alberta,<br />

Canada.<br />

EILOUTI, B. and A. ALJOKHADAR. 2007. A Computer-Aided Rule-Based Mamluk madrasa Plan<br />

Generator. Nexus Network Journal 9, 1: 31-58.<br />

GIPS J. 1975. Shape Grammars and their Uses. Basel: Birkhäuser.<br />

GLASSIE, H. 1975. A Structural Analysis of Historical Architecture. Knoxville: University of<br />

Tennessee Press.<br />

HIMMO, B. 1995. Geometry Working out Mamluk Architectural Designs: Case Study: Sabil Qaytbay<br />

in Holy Jerusalem. Master Thesis, University of Jordan, Jordan.<br />

The Islamic Methodology for the Architectural and Urban Design. 1991. Proceedings of the 4th<br />

seminar, Rabat. Morocco Organization of Islamic Capitals and Cities.<br />

KNIGHT T.W. 1999. Shape grammars: six types. Environment and Planning ‘B’: Planning and<br />

Design 26: 15-31.<br />

MARCH L. 1999. Architectonics of proportion: historical and mathematical grounds. Environment<br />

and Planning ‘B’: Planning and Design 26: 447-454.<br />

MICHEL, G., 1996. Architecture of the Islamic World: Its History and Social Meaning. London:<br />

Thames and Hudson.<br />

MITCHELL W.J. 1986. Formal representations: a foundation of computer aided architectural design.<br />

Environment and Planning ‘B’: Planning and Design 13: 33-162.<br />

OSMAN M.S. 1998. Shape grammars: simplicity to complexity. Paper presented in University of East<br />

London. (http://www.ceca.uel.ac.uk/cad/student_work/msc/ian/shape.html, accessed 1 Oct<br />

2003).<br />

Organization of Islamic Capitals and Cities. 1990. Workshop of Architectural and Urban Design<br />

Fundamentals in Islamic Periods, Islamic Architecture Heritage Conservation Center, Cairo,<br />

Egypt 1990.<br />

PARKER, R. 1985. Islamic Monuments in Cairo. 3rd ed. Cairo: The American University in Cairo<br />

Press.<br />

SERAGELDIN, R. 1988. Educational Workshop. In Places of Public Gathering in Islam. Proceedings of<br />

Seminar Five, Architectural Transformations in the Islamic World. Amman, Jordan: The Aga<br />

Khan Foundation for Architecture.<br />

STEADMAN J.P. 1983. Architectural Morphology: An Introduction to the Geometry of Building<br />

Plans. London: Pion.<br />

STIERLIN, Henri 1984. Great Civilizations: The Cultural History of the Arabs. London: Aurum Press.<br />

———. 1996. Islam: Early Architecture from Baghdad to Cordoba. Taschen Publishers.<br />

STINY G. and MITCHELL W. J. 1980. The grammar of paradise: on the generation of Mughul<br />

gardens. Environment and Planning ‘B’: Planning and Design 7: 209-226.<br />

STINY G. 1980. Introduction to shape and shape grammars. Environment and Planning ‘B’: Planning<br />

and Design 7: 343-351.<br />

WILLIAMS, John Alden. 1984. Urbanization and Monument Construction in Mamluk Cairo. In<br />

Muqarnas II: An Annual on Islamic Art and Architecture. Oleg Garbar (ed). New Haven: Yale<br />

University Press.<br />

28 B. EILOUTI and A. AL-JOKHADAR – A Generative System for Mamluk Madrasa Form-Making

About the authors<br />

Buthayna H. Eilouti is Assistant Professor and Assistant Dean at the Faculty of Engineering in<br />

Jordan University of Science and Technology. She earned a Ph.D., M.Sc. and M.Arch. degrees in<br />

Architecture from the University of Michigan, Ann Arbor, USA. Her research interests include<br />

Computer Applications in Architecture, Design Mathematics and Computing, Design Pedagogy,<br />

Visual Studies, Shape Grammar, Information Visualization, and Islamic Architecture.<br />

Amer Al-Jokhadar is a Part-Time Lecturer at the College of Architecture and Design in German-<br />

Jordanian University in Amman, and Architect in TURATH: Architectural Design Office. He<br />

earned M.Sc. and B.Sc. in Architectural Engineering from Jordan University of Science and<br />

Technology. His research interests include: Shape Grammar, Mathematics of Architecture, Computer<br />

Applications in Architecture, Islamic Architecture, Heritage Conservation and Management, Design<br />

Methods.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 29

Buthayna H. Eilouti<br />

Department of Architectural Engineering<br />

Jordan University of Science and<br />

Technology<br />

POB 3030<br />

Irbid 22110, JORDAN<br />

buthayna@umich.edu<br />

Amer M. I. Al-Jokhadar<br />

College of Architecture & Design<br />

German-Jordanian University<br />

Amman, JORDAN<br />

TURATH: Heritage Conservation<br />

Management and Environmental Design<br />

Consultants<br />

P.O.Box 402, Amman, 11118 JORDAN<br />

amerjokh@hotmail.com<br />

amerjokh@gmail.com<br />

Research<br />

A Computer-Aided Rule-Based<br />

Mamluk Madrasa Plan Generator<br />

Abstract. A computer-aided rule-based framework that<br />

restructures the unstructured information embedded in<br />

precedent designs is introduced. Based on a deductive<br />

analysis of a corpus of sixteen case studies from<br />

Mamluk architecture, the framework is represented as a<br />

generative system that establishes systematic links<br />

between the form of a case study, its visual properties,<br />

its composition syntax and the processes underlying its<br />

design. The system thus formulated contributes to the<br />

areas of design research and practice with a theoretical<br />

construct about design logic, an interactive<br />

computerized plan generator and a combination of a<br />

top-down approach for case study analysis and a<br />

bottom-up methodology for the derivation of artifacts.<br />

Keywords: Computer-aided design, logic of design,<br />

visual reasoning, precedent-based design, design<br />

process, mamluk architecture, shape grammar<br />

1 Introduction<br />

Design involves multiple activities and tasks that are based on reusing past design<br />

solutions. Design knowledge gained from studying precedents plays a significant role in<br />

the pre-design and precept reasoning stages. This is due to the fact that previous<br />

experiences help in understanding new situations and in casting older solutions into new<br />

problems, at least in the early stages of the problem-solving process. Precedent-based design<br />

experience is used when performing different tasks, whether they involve routine activities<br />

or require creative contributions. A process of interpolation and matching is conducted for<br />

the routine activities of comparison and exclusion in the processing of the creative ones.<br />

This is also the case in architectural design, where many problem-solving tasks are based on<br />

modifying past solutions. In fact, a major part of the pre-design investigation in<br />

architecture is concerned with studying case studies (precedents) that are related to the<br />

problem at hand. Multiple layers of information can be inferred from case studies to form a<br />

point of departure for new designs.<br />

Starting with a precedent-based design model can enhance the solution of a new design<br />

problem by taking as a point of departure a case as a whole or a combination of selected<br />

parts or features of different cases. In order to be of practical use, information inferred from<br />

precedents has to be represented in a meaningful way so as to enable designers to<br />

restructure and reassemble the extracted data for the generation of a new design. Precedentbased<br />

models can be represented in many forms. The one that will be emphasized in this<br />

paper is the Rule-Based Design (RBD) model, which Tzonis and White [1994, 20] called<br />

the Principle-Based Reasoning model. RBD provides dynamic and flexible solutions since<br />

many alternatives can be derived by applying a combination of rules (selected from a finite<br />

set) on a small set of basic shapes. In RBDs, data extracted from a group of precedents is<br />

Nexus Network Journal 9 (2007) 31-58 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 31<br />

1590-5896/07/010031-28 DOI 10.1007/S00004-006-0028-4<br />

© Kim Williams Books, Turin

formulated as concise productive rules and defined vocabulary elements by using a topdown<br />

approach. In this approach, a design problem is studied as a whole unit to derive the<br />

basic constituent elements and relationships underlying its structure. In this regard, RDB<br />

models can be associated with Problem-Based Learning techniques [Dochy et al. 2005] to<br />

strengthen their pedagogical power. In Problem-Based Learning settings, a whole problem<br />

is introduced, investigated and analyzed to allow learners to derive all the relevant basic data<br />

and details. For scope limitations, issues of Problem-Based Learning associations and<br />

impacts will be discussed in a separate paper.<br />

The main hypothesis of this study is that the logic of form making and style definition<br />

can be represented as a finite set of generative design rules and shape compositional<br />

principles. Thus, information embedded in precedents can be represented by a finite set of<br />

planning procedural rules and parameters for aesthetic articulation.<br />

The main scope of this research is to develop a computer-aided precedent-based system<br />

that can describe, analyze and generate two-dimensional representations of architectural<br />

design by applying productive rules of composition. The system is implemented on the<br />

floor plans of educational buildings (madrasas) in Mamluk architecture as case studies.<br />

The main goal of the study is to formulate a structured and systematic framework for<br />

analyzing the form of historical artifacts and for generating emergent designs based on the<br />

morphological analysis of precedents. The main objective is to restructure the originally<br />

unstructured or weakly-structured information embedded in precedents in the form of<br />

clearly formulated rules to make the information more usable and recyclable.<br />

The rule-based design framework developed in this paper is expected to contribute to<br />

the theory, practice and teaching of design. Based on a structuralist view 1 of the<br />

morphology of precedents [Caws 1988], the framework proposes a theoretical construct<br />

that encompasses a system of compositional and procedural rules for form generation. It<br />

enhances the body of knowledge in design areas by means of an externalized and explicit<br />

method of form-making and an incremental process for architectural composition, and by<br />

an applicable system for restructuring the weakly-structured information that is embedded<br />

in precedent designs. Furthermore, the paper emphasizes the aesthetic values of Mamluk<br />

architecture by systematically analyzing its morphology. For design practice, within the<br />

scope of this research, the use of the framework can help designers to visualize and evaluate<br />

several solutions of a given design problem, particularly in two-dimensional representations,<br />

and then to select one or a combination of these designs to develop. In addition, designers<br />

can automatically and systematically generate designs based on a knowledge base developed<br />

from the study of selected precedents. Regarding the pedagogical significance, applications<br />

in analysis and synthesis of basic components of compositions have important implications<br />

for design instructors who want to communicate the principles of visual composition and<br />

guidelines for the design process. In addition, the proposed framework represents an<br />

efficient way of learning about styles or languages of designs, especially about their<br />

compositional aspects. A language paradigm for form generation can also reveal general<br />

design strategies that students can learn from and use in solving their own design problems.<br />

By applying a bottom-up approach of rule search and matching to initial shapes, a student<br />

may better understand the form-making process and its associations with style definition.<br />

The research introduces a framework for a set of case studies, the morphology of which<br />

did not receive enough attention in the areas of the systematic analysis and synthesis, and<br />

32 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

the explicit articulation of the processes of their form-making. The precedents are selected<br />

from Mamluk architecture and are widely considered beautiful representative exemplars of<br />

Islamic architecture. The formulated framework proposes an integrated system that consists<br />

of a multi-layer structure and multi-phase sequence for deriving floor plans of Mamluk<br />

architecture.<br />

2 Background<br />

Most previous efforts in the area where precedent-based and rule-based design studies<br />

intersect are represented in the form of shape grammars. Shape grammar [Gips 1975; Stiny<br />

1980, 1994; Flemming 1987] is a method of the description, analysis, representation,<br />

classification, and generation of the form of an artifact and its incremental design process.<br />

In most cases, shape grammars are associated with the analysis of precedents. The common<br />

factor in the precedents studied is usually their designer, their style or the similar temporal<br />

or geographical context in which they were designed. Design precedents are typically<br />

investigated in order to understand the geometrical composition of their appearance and<br />

the process of their production. Then, a group of possible rules and initial shapes is derived.<br />

There are two principle kinds of shape grammar: standard grammar and parametric<br />

grammar. While in standard grammars most attributes of shapes are constant, in parametric<br />

grammars they tend to be more flexible and variable. Because of its flexibility, parametric<br />

grammar is more practical and popular than the standard one. If the number of parameters<br />

in parametric grammars is small, designers can recognize and predict several options of<br />

designs, but if the number is large and the spatial relations are more complex, shape<br />

grammars become too complex to be applied manually. Computer implementations of<br />

shape grammars facilitates the search through complex grammars or large numbers of rules<br />

and parameters. The relationship between shape grammars and computer implementations<br />

can be described in the statement:<br />

Shape grammars naturally lend themselves to computer implementations:<br />

the computer handles the bookkeeping tasks (the representation and<br />

computer computation of shapes, rules, and grammars, and the presentation<br />

of correct design alternatives) and the designer specifies, explores, develops<br />

design languages, and selects alternatives [Tapia 1999, 1].<br />

The concept of being able to derive new instances of a given style is certainly interesting<br />

as a way of extending the scope of a style by adding imaginary members that have the same<br />

compositional attributes. Such a derivation is based on the understanding of the formal<br />

patterns and compositional principles underlying the appearance of designed objects that<br />

belongs to that style. Consequently, it is fundamental to formulate the rules of a grammar<br />

in a flexible way so that the unexpected elements of design are allowed to emerge.<br />

In order to automate and implement shape grammars in an exploratory way, it is<br />

essential to formulate rules in an algorithmic, parametric and interactive format so when<br />

they are fed into the computer, they generate multiple alternatives of a design. The derived<br />

designs present possibilities that may inspire the designer. The rules should be based on the<br />

conceptual as well as the formal components of a design, and on the relational,<br />

morphological, and topological associations among all components. Computer<br />

implementations of shape grammars can function as educational tools for demonstrating<br />

the range and power of the grammars, and for illustrating the morphology and process of<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 33

designs. They enable students and designers to generate, analyze, evaluate, and select more<br />

rapidly and easily from among the various design alternatives that grammars generate<br />

without having to deal with the technicalities of grammar development.<br />

In spite of their theoretical appeal, shape grammars did not receive enough attention in<br />

the area of their automation. This is most probably the result of the relative complexity of<br />

their underlying algorithms and the difficulty of developing an integrated system for shape<br />

analysis and synthesis [Tapia 1999].<br />

According to Gips [2000], it is possible to identify four major types of the computer<br />

implementation of shape grammars. These are:<br />

1. A shape interpreter. In this type, a user defines a shape grammar in the computer,<br />

and the program generates shapes in the given language, either automatically or<br />

guided interactively by the user.<br />

2. A parsing program. A program of this type is given a shape grammar and a shape.<br />

The program determines if the shape belongs to the language generated by the<br />

grammar and, if so, gives the sequence of rules that produces the shape. This type<br />

focuses on the analysis and search aspects rather than the design generation issues.<br />

3. An inference program. In this type, the program is fed a coherent set of shapes of a<br />

given style, for which the computer automatically generates a shape grammar for<br />

the same style.<br />

4. A computer-aided shape grammar design program. A program of this type assists<br />

users in designing a shape grammar by providing sophisticated tools for rule<br />

construction and development.<br />

Some examples of the implementations that have been conducted on shape grammars<br />

are listed in Table 1. Gips’s interpreter [1975] enables the user to input a simple two-rule<br />

shape grammar; the program then generates two-dimensional shapes. Only polygonal<br />

vocabulary elements are allowed in this grammar. The program ignores the issue of<br />

detecting sub-shapes. Krishnamurti [1980, 1981-a, 1981-b] did pioneering work on<br />

developing data structures and algorithms for solving the sub-shape transformation and rule<br />

application problems for two-dimensional grammars. Using the Prolog programming<br />

language, Flemming [1987] developed a grammar for three-dimensional representations of<br />

the Queen Anne house, and Chase [1989] developed a general two-dimensional shape<br />

grammar system. Tapia’s GEdit [1999] is a general two-dimensional shape grammar<br />

interpreter that supports sub-shape detection and shape emergence. Agarwal and Cagan’s<br />

coffee maker grammar [1998] has been implemented using a JAVA program. A shape<br />

grammar interpreter was designed by Piazzalunga and Fitzhorn [1998] for threedimensional<br />

oblong manipulations using the LISP language. Also using a version of LISP,<br />

AutoLISP, Eilouti’s FormPro1 [2001] develops a universal grammar for three-dimensional<br />

architectural compositions with possible style variations for each resultant design. Duarte’s<br />

program [2005] implements a shape grammar for Malagueria house designs.<br />

In most previous efforts, computational problems related to encoding rules and their<br />

executions were focused on producing abstract shapes rather than architectural<br />

representations. They considered the layer of form processing that mostly concentrates on<br />

the construction grid and overall shapes of actual designs. Classification of components<br />

according to their functional and architectural articulations was in most cases ignored.<br />

34 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Furthermore, in most efforts, issues of designing the interactive and user-friendly interface<br />

of a program did not receive enough attention. Thus, most of these systems were not easy<br />

to use for nonprogrammers, novice users of shape grammars, or design practitioners.<br />

Table 1. A List of shape grammar computer implementations<br />

The computer-aided generative system presented in this paper is a multi-layer, multiphase<br />

shape interpreter that transforms a layout construction grid derived by rule<br />

applications in one phase into a full architectural plan with walls and openings in another<br />

phase. It is designed with a quickly-learned and easily-used interface. Furthermore, the<br />

stylistic prototypes of the program outcomes are developed from a valuable set of<br />

precedents that have not received enough attention of researchers, especially in the<br />

systematic and applied research areas. This set consists of Mamluk educational precedents.<br />

3 The theoretical framework for the rule-based generative system<br />

Grammatical inference can be defined as the task of logically inferring a finite set of rules<br />

from the systematic investigation of a coherent set of designed objects. In its architectural<br />

application, a building design or group of designs is defined. The designs are analyzed by<br />

decomposing them into a vocabulary, which represents the lexical level, and rules, which<br />

represent the syntactic level of their structure. Spatial relations, or arrangements of<br />

vocabulary elements in space, are identified in terms of the decompositions of the original<br />

designs. As a result of the re-synthesis of the decomposed elements, the shape grammar<br />

generates the original designs and possible new designs that did not exist before. New<br />

designs are generated by restructuring new combinations of the vocabulary elements in<br />

accordance with the inferred spatial relations, or by re-assembling the same elements using<br />

different rules or sequences.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 35

In order to develop a generative design system which applies rule-based assembly<br />

models, a foundation of a flexible parametric representation of rules and basic shapes must<br />

be formulated. The rules and initial shapes are inferred from a set of precedents, drawn in<br />

the present paper from Mamluk architecture. The study focuses on a set of educational<br />

building (madrasas) that belong to this style. Mamluk madrasas represent some of the most<br />

beautiful examples of the Islamic architecture. Despite their historical and aesthetic value,<br />

they have not received enough attention from researchers, especially in the areas of<br />

systematic analysis of their morphological structures or designs.<br />

This research examines a group of sixteen Mamluk madrasas from the different regions<br />

ruled during the Mamluk period. The case studies include thirteen precedents from Egypt<br />

(located in Cairo), one precedent from Syria (located in Aleppo), and two precedents from<br />

Palestine (located in Jerusalem). The study cases from Cairo are: al-Ashraf Barsbay, al-Kadi<br />

Zein al-Dien Yehya, al-Amir Kerkamas, al-Sultan Inal, al-Ghouri, Umm al-Sultan Sha’ban,<br />

Kani Bay Kara al-Remah, al-Sultan Kaytebay, and Abu-Baker Mezher Madrasas from the<br />

covered court type; and al-Thaher Barquq, al-Sultan Hasan, Serghtemetsh, and al-Sultan<br />

Qalawun Madrasas of the open court type. The study cases from Jerusalem are: the<br />

Tashtamar Madrasa from the covered court type and al-Baladiyya Madrasa of the open<br />

court type. The case from Aleppo is al-Saffaheyya Madrasa which is of the open court type.<br />

The morphological study of the case studies, as well as the spatial organization of the<br />

components is discussed in a separate paper [Eilouti and Al-Jokhadar 2007].<br />

As a result of the structuralist analysis (see note 1) to find the commonalities that<br />

underlie the cases, and the comparisons of the sixteen precedents, it is concluded that in<br />

order to reconstruct a floor plan of a Mamluk madrasa, five sets of rules need to be defined.<br />

The first set is the rules for schematic layout. Applied in a top-down, out-in approach<br />

where form evolves from the exterior-most abstracted shapes into the interior and detailed<br />

articulations, rules in this set are used to derive the shape of the exterior layout and its<br />

major organizational axes. The second set consists of the rules for the architectural<br />

components, which articulates the interior space and room shapes based on their functions<br />

and the logic of how the components are related within the floor plan diagram. Rules in the<br />

first two sets are schematic and are applied to establish the construction lines and major<br />

axes of orientation and symmetry. The third set includes rules for determining wall<br />

thicknesses. These assign a thickness for the construction lines generated through the layout<br />

and architectural component rules. Thus, rules in the third set transform the schematic<br />

diagram generated by the first two sets into an architectural representation of the floor plan<br />

at hand. The fourth set consists of rules for openings. These add shapes for door and<br />

window openings to the floor plan layout generated by the previous three sets of rules. The<br />

fifth set includes the rules for termination. These conclude the process of floor plan<br />

generation.<br />

The five sets of rules and the information they represent are illustrated in Table 2. Each<br />

set represents a phase in the derivation sequence. The generation process starts with phase<br />

one, cycles between the second, third and fourth phases as needed, and finally concludes<br />

with the fifth phase, which terminates steps of the process. Rules in each phase are listed<br />

with their count and information as follows:<br />

The formulated grammar consists of 93 rules distributed as:<br />

36 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Rule type<br />

Number of Rules<br />

Rules for generating schematic external layout shapes 12<br />

Rules for generating architectural components 59<br />

Rules for determining wall thickness 9<br />

Rules for generating openings 9<br />

Termination rules 4<br />

The rules in each category are listed in Table 2.<br />

Table 2: The rule system for the derivation of Mamluk madrasa precedents<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 37

The Mamluk madrasa grammatical rule numbering system, and the initial and the<br />

exterior layout shapes are illustrated a separate paper [Eilouti and Al-Jokhadar 2007].<br />

As an example of how the formulated rules restructure the information embedded in the<br />

case studies, rules (AC-5-D) and (AC-6-M) specify that there are transitional spaces<br />

between the courtyard and two Iwans. In addition, Rule (AC-7-S) specifies the dimensions<br />

of the courtyard (V B ) and (H B ). The proportion between (V B ) and (H B ) is 1:1, while the<br />

proportion between the courtyard and the total interior space is around 1:2.54, 1:2.30, or<br />

1:2.22.<br />

Rules (AC-8-S) and (AC-9-S) determine the dimensions of the two Iwans (H C : V C ) and<br />

(H D : V D ). All of these spaces are symmetrical about the main axis Y 2 – Y 2 . Fig. 1<br />

illustrates three of these rules and their associated symbols. These rules, namely (AC-7-S),<br />

(AC-8-S) and (AC-9-S), are used for scaling two of the major components of madrasa plan<br />

morphology, the courtyard and the Iwan components.<br />

As an example of the information extracted from the floor plans of the sixteen<br />

precedents, the mathematical relations and ratios that govern the courtyard geometry of the<br />

case studies are illustrated in Table 3. The H and V values in the table denotes the<br />

horizontal and vertical dimensions of the court B. B represents the area of the court, and A<br />

represents the area of the court and the spine of the Iwans. B:All represents the area of the<br />

court compared to the total area of the madrasa at hand. X,Y signifies the origin point<br />

which is usually the centre of courtyard. These proportional and locational values of shape<br />

are translated into parametric assignment for the rules in the generative system structure.<br />

A numerical analysis similar to that illustrated in Table 3 is conducted on all<br />

architectural components of the case studies.<br />

The five-fold framework outlined above is developed into a computer implementation,<br />

which is designed to be viable for practical design problems. The computerized framework<br />

is designed to enable designers to understand existing designs and to generate new forms<br />

that have the same style of the studied precedents.<br />

The computerized version of the generative system reflects the multi-layered multiphased<br />

characteristics of the rule structure in order to permit a rapid generation of Mamluk<br />

madrasa prototypes through an incremental and lucid step-by-step sequence of selecting<br />

shapes, operators, transformations, parameter assignment, and rule application.<br />

The method used to develop the generative system for the Mamluk madrasa design can<br />

be applied to any set of precedents and can be extended to allow experimentation and<br />

exchange between designs and styles.<br />

38 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Fig. 1. Rules for proportional scaling of courtyard and Iwans<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 39

Table 3. The parameters that control the shapes of the courtyards of Mamluk madrasas<br />

4 The program interface<br />

In general, it is widely assumed that people who use shape grammars tend to be visual<br />

thinkers, and those who program and implement computer codes are symbolic thinkers.<br />

People who think well both visually and symbolically seem be quite rare [Gips 2000]. This<br />

may explain the insufficiency of computer implementations of shape grammar systems and<br />

the problem of the interface design in the existing implementations.<br />

The issue of successful user design interface is significant in the design of computer<br />

implementations for shape grammars. Interface issues were addressed in a comprehensive<br />

way in Tapia [1996; 1999]. Tapia emphasizes the importance of improved computational<br />

machinery for a general two-dimensional shape grammar interpreter, along with a simple,<br />

intuitive, visual interface. The user of a computer-aided rule-based interpreter program<br />

needs to learn both about rule structure and how to use the program. To facilitate this, the<br />

program design should be transparent. Certainly it should make it easier to use the<br />

program than to try out rule application by hand [Gips 2000].<br />

40 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

To generate or reconstruct a floor plan for Mamluk madrasas, the program starts by<br />

parsing and classifying the main vocabulary elements, their relationships and possible<br />

transformations. The overall flowchart of the rule development system is shown in fig. 2.<br />

Because of the large number of rules, the program commands are designed in phases and<br />

are decomposed into sequences of rules. The codes of the generative system are written in<br />

the AutoLISP language for the AutoCAD operating environment. This saves time in<br />

rewriting some existing programming routines and is very suitable for graphic applications.<br />

The interface of the program described here consists of five main menus in addition to<br />

the typical file management, editing, viewing and support window-compatible menus. The<br />

five menus represent the five phases of the derivation process described above. Each phase<br />

reveals the multi-layered structure of the madrasa composition at that stage. The menus are<br />

designed in a simple and clear format. The menus are:<br />

1. “Schematic Layout” rules. This menu specifies seven functions: the first six<br />

illustrate the six different types of layout for generating the exterior layout<br />

shapes of madrasa floor plans. Each layout has two rules. The first determines<br />

the proportions of the sides of the shape, and the second determines the angles<br />

between each two sides. The seventh and last function in this menu defines the<br />

centre of the exterior layout with reference to the interior court.<br />

2. “Architectural Components” rules. This menu has nine submenus:<br />

(1) Interior spaces. This submenu includes three rules for inner space<br />

organization.<br />

(2) Courtyard and Iwans. This submenu includes twelve rules for the open<br />

and semi-open space articulation within the exterior layout shape.<br />

(3) Mihrab. This submenu includes three rules for locating the niche in the<br />

southern Iwan wall.<br />

(4) Articulating interior spaces. This has three rules for shaping the minor<br />

inner spaces.<br />

(5) Sadla. This submenu includes one rule for forming the geometry of the<br />

secondary Iwan.<br />

(6) Tomb. This submenu branches into four submenus: tomb #1 (with four<br />

rules); tomb #2 (with four rules); tomb #3 (with four rules); and tomb<br />

#4 (with one rule).<br />

(7) Ablution space. This submenu includes two rules for the articulation of<br />

the ablution room.<br />

(8) Cells around the courtyard. This submenu includes four rules for the<br />

articulation of the supportive cell geometry.<br />

(9) Main entrance and derka. This submenu branches into six submenus:<br />

entrance #1 (with three rules); entrance #2 (with three rules); entrance<br />

#3 (with three rules); entrance #4 (with three rules); entrance #5 (with<br />

three rules); and entrance #6 (with three rules).<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 41

Fig. 2. A flow chart illustrating the process of developing the rule-based model<br />

42 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Fig. 3. The hierarchical organization of the rules of the Mamluk Madrasa program interface<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 43

3. “Walls Thicknesses” rules. The main function for this menu is determining the<br />

thicknesses of exterior and interior walls of madrasas. For applying thickness to<br />

designated walls, the menu offers nine rules.<br />

4. “Openings” rules. The main function for this menu is assigning the doors and<br />

windows for the plan of the madrasa. This is done through specifying whether<br />

the tomb is located on the right or left side of the Qibla-Iwan. There are nine<br />

rules under this menu.<br />

5. “Termination” rules. These will erase the labels assigned to shapes through<br />

different stages for developing the shape grammars. This menu has four rules.<br />

All of the five main menus and the additional submenus are illustrated in the<br />

hierarchical organization diagram shown in fig. 3. The information described by the rules<br />

is listed in Table 2.<br />

The structure of the aforementioned five menus is shown in images captured from the<br />

screen, as illustrated in figs. 4 to 8.<br />

Fig. 4. A screen image of the schematic layout menu<br />

44 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Fig. 5. A screen image of the architectural component menu<br />

Fig. 6. A screen image of the wall thickness menu<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 45

Fig. 7. A screen image of the openings menu<br />

Fig. 8. A screen image of the termination menu<br />

46 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

5 Program testing and discussion<br />

The Case of al-Ashraf Barsbay Madrasa in Cairo is used to demonstrate the applicability<br />

of the program presented in the previous sections. This case represents one of the sixteen<br />

precedents analyzed. It exhibits most of the features and components of the generative<br />

system. The generation process starts with the basic layout shape shown in fig. 9. The shape<br />

of the exterior layout is selected from the set of initial shapes offered by the schematic<br />

layout menu.<br />

The plan generation process continues by applying rules from the architectural<br />

component menu to add the overall layout shape of the interior spaces and to assign<br />

numerical values for the parameters required by the rule prompts. The result of the interior<br />

space rule application process is illustrated in fig. 10.<br />

Other architectural components, such as the courtyards, iwans, mihrab, minbar, sadla,<br />

tomb, ablution space, cells around the courtyard, the main entrance, derka and the Sheikh’s<br />

house, are also added by using the architectural components menu. This set of architectural<br />

component configuration and articulation steps of the derivation process is illustrated in<br />

figs 11-17.<br />

Fig. 9. The exterior layout of al-Ashraf Barsbay Madrasa after applying rules from the schematic<br />

layout menu<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 47

Fig. 10. The overall layout of the interior spaces of al-Ashraf Barsbay Madrasa<br />

Fig. 11. The addition of the courtyards and iwans for al-Ashraf Barsbay Madrasa<br />

48 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Fig. 12. Generating the mihrab for al-Ashraf Barsbay Madrasa<br />

Fig. 13. Generating minbar and sadla for al-Ashraf Barsbay Madrasa<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 49

Fig. 14. Generating the tomb for al-Ashraf Barsbay Madrasa<br />

Fig. 15. Generating the ablution space for al-Ashraf Barsbay Madrasa<br />

50 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Fig. 16. Generating cells around the courtyard for al-Ashraf Barsbay Madrasa<br />

Fig. 17. Generating the main entrance, derka, and the Sheikh’s house for al-Ashraf Barsbay Madrasa<br />

After the definition of the main architectural components of the madrasa, the door and<br />

window openings are added by using the opening menu. The plan with the openings is<br />

shown in Fig. 18.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 51

Fig. 18. Generating openings (doors and windows) for al-Ashraf Barsbay Madrasa<br />

Fig. 19. The overall layout of al-Ashraf Barsbay Madrasa after applying rules for the assignment of<br />

wall thicknesses<br />

52 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

Fig. 20. A new alternative for the final layout of al-Ashraf Barsbay Madrasa after changing the exterior<br />

layout, location of the entrance, derka, and the tomb (in front of qibla iwan)<br />

The penultimate step of the derivation process is the assignment of wall thicknesses to<br />

the lines derived so far. The assignment is conducted by applying the wall thickness menu<br />

commands. The result of this process, which transforms the construction line grid into the<br />

representation of the architectural plan, is shown in fig. 19.<br />

To explore the emergent component of the program, variations in the order of rule<br />

selection and application were experimented with. By transforming, combining, and<br />

replacing layout shapes and rules, the generated designs can be made more interesting. As<br />

an example, various alternative designs that rely on the same principles of Mamluk madrasa<br />

design could be generated for al-Ashraf Barsbay Madrasa. One of these is shown in fig. 20,<br />

which shows the plan of the madrasa after changing the location of the main entrance, the<br />

derka and the tomb, and after varying the exterior layout shape.<br />

Applying the automated grammar program highlights and explains through repetition<br />

the numerical, relational, and morphological aspects of plan organization of Mamluk<br />

architecture, which are usually neglected in descriptions of the style. Mamluk madrasas<br />

built in Egypt, Syria and Palestine have many morphological similarities. As shown by the<br />

derived example, the location of some components, such as the courtyard and the main<br />

Iwans, represents a major factor in shaping the grammar of all architectural compositions of<br />

Mamluk madrasa architecture. In addition, visual principles such as proportion, symmetry,<br />

axis location, and rotation angles are shared by the case studies and are emphasized during<br />

the generation process.<br />

Knowledge of the geometric characteristics of the rudimentary vocabulary elements can<br />

be gained and enhanced by manipulating the exterior layout and interior space shapes that<br />

are offered by the menus for schematic layout and the architectural components.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 53

Knowledge of the scale, proportion, orientation, symmetry, rhythm, and other visual<br />

principles can be enhanced by the numerical assignment of the parameters that are required<br />

during the rule application process. The shape grammar of Mamluk madrasas is influenced<br />

by many visual principles. These include symmetry, unit-to-whole ratio, repetition,<br />

addition and subtraction, geometry and grid structure, hierarchy, complexity, and<br />

orientation of parts. The major principles are briefly described as follows:<br />

Asymmetric balance: As shown by the example derived, the typical symmetry that is<br />

associated with the sacred precedents is substituted by a concept of order that provides<br />

balance between parts which form a coherent whole. This concept was achieved through<br />

the use of multiple axes of various directions to control all topological relationships of the<br />

vocabulary of Mamluk madrasa components. Balance is achieved by geometry when a<br />

dominant form, such as the dome on the mausoleum, balances a massive cuboid that is<br />

dominant by its size. Although asymmetry is the general characteristic, Mamluk<br />

architecture exhibit some levels of symmetry on the organization of the internal layout. The<br />

precedents studied reveal central symmetry in which major axes meet in a focal point that is<br />

typically located at the centre of the courtyard and controls the layout of the Iwans.<br />

However, symmetry in Mamluk madrasas was not perfect. It was applied to control the<br />

organization of some internal components of the floor plan rather than the overall exterior<br />

shape.<br />

Rhythm: Repetition, expressed as a rhythm of component distribution, can be found in<br />

many instances of Mamluk madrasa floor plans. It is expressed in the form of addition or<br />

division of a whole, or simply represented as a series of ratios. Elements of the plan were<br />

repeated by using the ratios of 1:1; or less frequently 1:2.57, 1:1.6 (approximately the<br />

golden section), or 1:3.14 (1: ).<br />

Orientation of elements: A major factor of coherence in Mamluk madrasa architecture<br />

depends on grouping elements which have the same function on the same orientation axis.<br />

The major axis of orientation is dictated by the angle of Qibla direction. Aagain, this is due<br />

to the connection between education and religion in Mamluk madrasa architecture.<br />

Hierarchy: Hierarchy was achieved in Mamluk madrasas by ordering primary components<br />

(such as the main courtyard) in the most central position and the secondary elements (such<br />

as cells around courtyard) in the less dominant locations.<br />

Simplicity vs. Complexity: Although all interior spaces are simply shaped, the overall layout<br />

has a complex form. This complexity refers to two features of the layout which prevent the<br />

fragmentation of the secondary components:<br />

1. The change of angle was used as a planning technique to create a new axis of<br />

orientation to organize the minor vocabulary elements. While the primary<br />

components were directed to the Qibla direction, the secondary ones were<br />

made more dynamic by changing their directions to distinguish them visually.<br />

2. The large central space acts as a reference point or datum line for all other plan<br />

components.<br />

Understanding the morphology and the design process of individual Mamluk buildings<br />

may enhance the explanation of town and city growth in Islamic cultures. By projecting<br />

component structure and relations in a Mamluk building on various buildings and their<br />

54 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

interrelationships in town planning, new layers of interpretation can be added. For<br />

example, the previous characteristics such as space hierarchy and angle manipulation can be<br />

observed as planning trechniques in some Islamic cities.<br />

The example derived in this section illustrates the time-saving advantage of the program.<br />

For example, the manual generation of the floor plan required eight hours as opposed to<br />

one hour required for the computer-aided derivation of the same plan. The computer<br />

implementation presented in this paper represents a tool for designing an accurate and<br />

multi-layered madrasa floor plan.<br />

Using the program conveys a message to the user that the process is as important as, if<br />

not more important than, the final product. It facilitates the exploration of the design<br />

process and product morphology analysis to better understand the beauty underlying its<br />

design. The developed grammar with its systemized components (vocabularies,<br />

mathematical and topological relations, rules, and initial shapes) enhances knowledge of<br />

design science through an algorithm-based framework that performs instantiation,<br />

transformation, and combination, as well as set operations of shapes and through<br />

connecting style attributes with principles of visual composition.<br />

Although the generative system has good predictive, derivative, and descriptive powers,<br />

it has limitations when it comes to explaining the historical, social, and other symbolic and<br />

semantic considerations in the architectural composition. A future extension of the<br />

program may link the historical and social issues to the syntactic grammar developed in this<br />

paper in the form of attributed rules that augment semantic interpretations of the<br />

grammatical formalisms. Such augmentation can take the form of conditional or contextsensitive<br />

rule application. As such, restrictions can be added to guide the users about which<br />

rules to select at each step and in what context to apply them.<br />

6 Conclusion<br />

It has been shown in this paper that a Rule-Based Design (RBD) system can be<br />

constructed from the information inferred from case studies to guide the design process.<br />

Most of the design information embedded in precedent designs is not in a format that is<br />

viable for direct reuse. This paper has suggested a five-phase system of rules to enable<br />

restructuring and representing of the unstructured information embedded in precedents in<br />

the form of reproductive and recursive rules. Such a representation helps in the<br />

regeneration of existing precedents and in the generation of new designs that belong to the<br />

same stylistic prototypes. The system makes it possible to experiment with various<br />

combinations of rule application to explore new and unexpected alternatives.<br />

After the theoretical formulation of the precedent-based generative system, it is<br />

translated into a computer-aided program. The program facilitates the automatic<br />

exploration of the aesthetic values of existing Mamluk precedents, as well as the generation<br />

of emergent examples of the same style.<br />

This paper claims that a large portion of the rules used to define a style are based on the<br />

logic of form-making. The system developed supports this claim by articulating two types<br />

of form-making rules. These include the planning-based rules and the aesthetic-related<br />

parameters and constraints. The first type is concerned with the procedural aspects and the<br />

compositional principles and is expressed as a set of organizational axes and focal points of<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 55

form-making and is translated into direct rules. The second focuses on the numerical<br />

assignment of the dynamic parameters.<br />

The research has shown that an integrated system that is based on a combination of a<br />

top-down approach of extracting information from precedents and a bottom-up approach<br />

of representing the extracted data as a set of rules can become a powerful tool for form<br />

analysis and derivation. The indeterministic nature of the system, coupled with its<br />

parametric structure, makes it flexible and dynamic enough to come up with unexpected<br />

emergent designs. The theoretical framework as well as the computerized program can be<br />

added to the design toolbox to communicate form structure, design process, principles of<br />

aesthetic composition and stylistic alphabet, and syntax of architectural design. Limitations<br />

of the system include the lack of representation of the social, historical, symbolic, and<br />

semantic issues in the form-making process.<br />

A future extension of this research may focus on studying sets of precedents of different<br />

styles and developing a multi-style automated generative system and style editor that<br />

compares, evaluates, analyzes, and produces architectural designs of various styles. Another<br />

extension might take into consideration the morphology of façades or the threedimensional<br />

spatial organizations of the Mamluk precedents as well as their impacts on<br />

plan composition. A comparative study could also be conducted to find commonalities or<br />

correspondences between the grammar of Mamluk madrasas and that of other madrasas<br />

(such as Western madrasas). Furthermore, connections between this research and Problem-<br />

Based Learning or E-Learning settings could be emphasized in future research.<br />

Notes<br />

1. According to structuralism [Caws 1988], objects exist in groups that collectively exhibit<br />

commonalities in their attributes. Such objects can be grouped into systems which can be<br />

defined recursively. Consequently, to understand systems it is necessary to investigate the<br />

internal as well as the external relationships of the system components and the structures that<br />

underlie their grouping.<br />

References<br />

AGARWAL, M., and J. CAGAN. 1998. A blend of different tastes: the language of coffee makers.<br />

Environment and Planning ‘B’: Planning and Design 25: 205-226.<br />

AL-JOKHADAR, A. 2004. Shape Grammar: An Analytical Study of Architectural Composition Using<br />

Algorithms and Computer Formalisms (The Morphology of Educational Buildings in Mamluk<br />

Architecture). Unpublished Master Thesis, Department of Architecture, Jordan University of<br />

Science and Technology, Irbid, Jordan.<br />

CAWS, P. 1988. Structuralism: The Art of the Intelligible. Atlantic Highlands: Humanities Press<br />

International.<br />

CHASE, S.C. 1989. Shapes and shape grammars: from mathematical model to computer<br />

implementation. Environment and Planning ‘B’: Planning and Design 16: 215-242.<br />

DOCHY, F., M. SEGERS, P. VAN DEN BOSSCHE and K. STRUYVEN. 2005. Students’ Perceptions of a<br />

Problem-Based Learning, Environment, Learning Environments Research 8: 41–66.<br />

DUARTE, J. P. 2005. Towards the mass customization of housing: the grammar of Siza’s houses at<br />

Malagueira. Environment and Planning ‘B’: Planning and Design 32(3): 347 – 380.<br />

EILOUTI, B. 2001. Towards a Form Processor: A Framework for Architectural Form Derivation and<br />

Analysis Using a Formal Language Analogy. Ph.D. Dissertation, University of Michigan.<br />

EILOUTI, B. and A. AL-JOKHADAR. 2007. A Generative System for Mamluk Madrasa Form-Making.<br />

Nexus Network Journal 9, 1: 7-30.<br />

56 B. EILOUTI and A. AL-JOKHADAR – A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator

FLEMMING, U. 1987. More than the sum of its parts: the grammar of Queen Anne houses<br />

Environment and Planning B: Planning and Design 14: 323-350.<br />

GIPS, J. 1975. Shape Grammars and their Uses. Basel: Birkhäuser.<br />

———. 2000. Computer Implementation of Shape Grammars. Boston: Computer Science<br />

Department, Boston College.<br />

GLASSIE, H. 1975. A Structural Analysis of Historical Architecture. Knoxville: University of<br />

Tennessee Press.<br />

The Islamic Methodology for the Architectural and Urban Design. 1991. Proceedings of the 4th<br />

seminar, Rabat, Morocco. Organization of Islamic Capitals and Cities.<br />

KONING, H. and J. EISENBERG. 1981. The language of the prairie: Frank Lloyd Wright's prairie<br />

houses. Environment and Planning: B 8: 295-323.<br />

KRISHNAMURTI, R. 1980. The arithmetic of shapes. Environment and Planning ‘B’: Planning and<br />

Design 7: 463-484.<br />

———. 1981a. The construction of shapes, Environment and Planning ‘B’: Planning and Design 8:<br />

5-40.<br />

———. 1981b. SGI: A Shape Grammar Interpreter. Research Report, Centre for Configurational<br />

Studies, The Open University, Milton Keynes, UK.<br />

———. 1992a. The maximal representation of a shape. Environment and Planning ‘B’: Planning<br />

and Design 19: 267-288.<br />

———. 1992b. The arithmetic of maximal planes. Environment and Planning ‘B’: Planning and<br />

Design 19: 431-464.<br />

KRISHNAMURTI, R. and C. GIRAUD. 1986. Towards a shape editor: the implementation of a shape<br />

generation system. Environment and Planning ‘B’: Planning and Design 13: 391-403.<br />

OSMAN, M.S. 1998. Shape grammars: simplicity to complexity. Paper presented in University of East<br />

London, London.<br />

http://ceca.uel.ac.uk/cad/student_work/msc/ian/shape.html)<br />

PIAZZALUNGA, U. and P.I. FITZHORN. 1998. Note on a three–dimensional shape grammar<br />

interpreter, Environment and Planning B: Planning and Design 25: 11–33.<br />

STINY, G. 1975. Pictorial and Formal Aspects of Shape and Shape Grammars. Basel: Birkhäuser.<br />

———. 1980. Introduction to shape and shape grammars, Environment and Planning ‘B’: Planning<br />

and Design 7: 343-351.<br />

———. 1994. Shape rules: closure, continuity and emergence. Environment and Planning ‘B’:<br />

Planning and Design, vol. 21, Pion Publication, Great Britain, pp. s 49- s 78, 1994<br />

TAPIA, M. A. 1996. From Shape to Style. Shape Grammars: Issues in Representation and<br />

Computation, Presentation and Selection. Ph.D. Dissertation, Department of Computer Science,<br />

University of Toronto, Toronto.<br />

———. 1999. A visual implementation of a shape grammar system. Environment and Planning ‘B’:<br />

Planning and Design 26: 59-73.<br />

TZONIS, A. and I. WHITE, eds. 1994. Automation Based Creative Design: Research and Perspectives,<br />

London: Elsevier Science.<br />

About the authors<br />

Buthayna H. Eilouti is Assistant Professor and Assistant Dean at the Faculty of Engineering in Jordan<br />

University of Science and Technology. She earned a Ph.D., M.Sc. and M.Arch. degrees in<br />

Architecture from the University of Michigan, Ann Arbor, USA. Her research interests include<br />

Computer Applications in Architecture, Design Mathematics and Computing, Design Pedagogy,<br />

Visual Studies, Shape Grammar, Information Visualization, and Islamic Architecture.<br />

Amer Al-Jokhadar is a Part-Time Lecturer at the College of Architecture and Design at the German-<br />

Jordanian University in Amman, and an architect in TURATH: Architectural Design Office. He<br />

earned M.Sc. and B.Sc. degrees in Architectural Engineering from Jordan University of Science and<br />

Technology. His research interests include Shape Grammar, Mathematics of Architecture, Computer<br />

Applications in Architecture, Islamic Architecture, Heritage Conservation and Management, Design<br />

Methods.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 57

Dirk Huylebrouck<br />

Department for Architecture<br />

Sint-Lucas<br />

Paleizenstraat 65-67<br />

1030 Brussels BELGIUM<br />

Huylebrouck@gmail.com<br />

Keywords: curve fitting,<br />

design analysis, Antoni Gaudí,<br />

linear algebra, generalised<br />

inverses, least squares,<br />

hyperboloid, catenary, parabola,<br />

golden number<br />

Research<br />

Curve Fitting in Architecture<br />

Abstract. It used to be popular to draw geometric figures on<br />

images of paintings or buildings, and to propose them as an<br />

“analysis” of the observed work, but the tradition lost some<br />

credit due to exaggerated (golden section) interpretations. So,<br />

how sure can an art or mathematics teacher be when he wants to<br />

propose the profile of a nuclear plant as an example of a<br />

hyperboloid, or proportions in paintings as an illustration of the<br />

presence of number series? Or, if Gaudi intended to show chain<br />

curves in his work, can the naked eye actually notice the<br />

difference between these curves and parabolas? The present paper<br />

suggests applying the “least squares method”, developed in<br />

celestial mechanics and since applied in various fields, to art and<br />

architecture, especially since modern software makes<br />

computational difficulties nonexistent. Some prefer jumping<br />

immediately to modern computer machinery for visual<br />

recognition, but such mathematical overkill may turn artistic<br />

minds further away from the (beloved!) tradition of geometric<br />

interpretations.<br />

Introduction<br />

Until about twenty years ago, it was common to draw all kinds of geometric figures on<br />

images of artworks and buildings. Usually, simple triangles, rectangles, pentagons, or circles<br />

sufficed, but sometimes more general mathematical figures were used, especially after<br />

fractals became trendy. Recognizing well-known curves and polygons was seen as a part of<br />

the “interpretation” of an architectural edifice or painting. Eventually, segments of lines<br />

occurred in certain proportions, among which the golden section surely was the most<br />

(in)famous. Diehards continue this tradition, though curve drawing has lost some credit in<br />

recent times, in particular due to some exaggerated interpretations of the golden section.<br />

Today, journals dealing with the nexus of mathematics and architecture tend to reject<br />

these “geometric readings in architecture”. Their point of view may be justified by several<br />

shortcomings of common research in the field:<br />

i. An architect may have had the intention of constructing a certain curve or<br />

surface, and even discuss these intentions in his plans, but for structural,<br />

technical or various practical reasons, the final realization may not conform to<br />

that intention.<br />

Example: a nuclear plant is said to have the shape of a hyperboloid, but engineers modify<br />

its top to reduce wind resistance. So, how close to a hyperbola is the final silhouette of the<br />

structure?<br />

ii.<br />

An artist may have used a certain proportion, consciously or unconsciously, so<br />

that when such a “hidden” proportion or curve is discovered, even the author<br />

of the artwork may dispute its use.<br />

Nexus Network Journal 9 (2007) 59-70 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 59<br />

1590-5896/07/010059-12 DOI 10.1007/S00004-006-0029-3<br />

© Kim Williams Books, Turin

Example: It happens that rational subdivisions are discovered in impressionist paintings.<br />

Some artists claim they intended only pure emotion, disliking art-school-like subdivisions<br />

such as the golden section. However, some critics can still demonstrate the presence (or<br />

absence) of this proportion using arguments similar to widely accepted interpretations of<br />

other artwork.<br />

iii. In the absence of accurate written sources about a work of art or an<br />

architectural design, how does one decide whether an interpretation is correct<br />

or not? Is it merely a matter of a subjective interpretation, a posteriori?<br />

Example: in a discussion about pointed arches, how does one decide if an arch was pointed<br />

or not, without being mislead by cultural penchants?<br />

Despite this unenthusiastic introduction, this author still believes there could be positive<br />

reasons for continuing the study of geometric views on art and architecture:<br />

i. An engineer may want to reveal structural properties of the building.<br />

Example: the difference between a parabola or chain curve when a cable supports a<br />

horizontal bridge, or only its own weight.<br />

ii.<br />

A mathematician may want to express what he finds beautiful in his preferred<br />

mathematical language and there is no reason why only literary minds should<br />

enjoy this privilege.<br />

Example: Horta was inspired by spirals he founds in plants. These spirals have a precise<br />

mathematical description, and though it is not certain that Horta intended to represent<br />

these curves, the mathematical interpretation may show a relationship with other sources of<br />

inspiration by other architects. If another architect was inspired by a nautilus shell, then<br />

both, in fact, had a similar inspiration, at least, as far as the mathematician is concerned.<br />

iii.<br />

An artist or architect may have been inspired by a curve or a proportion he<br />

finds elegant.<br />

Example: well-defined proportions may have been used to subdivide a canvas or to<br />

construct buildings. This can help to collect information on how the work was done, and,<br />

for instance, to determine the unit length used (non-standard foot, cubit).<br />

In the present paper, a similarity is proposed between these geometric studies in<br />

architecture and the history of (celestial) mechanics. It is suggested that the so-called “least<br />

squares method” developed in that field could be applied to examples in art as well. Of<br />

course, it can be argued that, unlike celestial mechanics with its involved applications, such<br />

a mathematical method would represent serious overkill with respect to the intended<br />

straightforward artistic applications. However, if this was true in the past, modern software<br />

allows slimming down computational aspects to some simple computer clicks.<br />

Campbell and Meyer’s account on curve fitting<br />

A well-known example in the history of curve-fitting is the story of how Carl Friedrich<br />

Gauss found a “lost planet”. Campbell and Meyer [1979] gave the following excellent<br />

account illustrating curve fitting, in the context of celestial mechanics and linear algebra.<br />

In January 1801, astronomer G. Piazzi briefly observed a “new planet”, Ceres, and<br />

astronomers tried, in vain, to relocate it for the rest of 1801. Only Gauss could correctly<br />

60 DIRK HUYLEBROUCK – Curve Fitting in Architecture

predict when and where to look for the lost planet but because he waited until 1809 to<br />

publish his theory, some accused him of sorcery.<br />

For the sake of exposition, Campbell and Meyer simplified the problem to an elliptical<br />

orbit of which four observations were made: (-1,1), (-1,2), (0,2), (1,1). In this example,<br />

obviously, there must have been some errors, of whatever kind (inaccurate observations or<br />

instruments, or mere bad luck), since two data correspond to -1. Still, the question now is<br />

to find the ellipse<br />

x<br />

a<br />

2<br />

2<br />

<br />

b<br />

y<br />

approximating the four data as closely as possible.<br />

2<br />

2<br />

1 ,<br />

Fig. 1. Campbell and Meyer’s example of finding the ellipse closest to four data points<br />

Putting b 1 =1/a 2 and b 2 =1/b 2 , implies the four data points (x i , y i ), i = 1, ...4, should<br />

make the error x i 2 b 1 + y i 2 b 2 – 1 minimal. In matrix notation, it means X.b-j should be<br />

minimal, where:<br />

x<br />

<br />

X = x<br />

x<br />

<br />

<br />

x<br />

2<br />

1<br />

2<br />

2<br />

2<br />

3<br />

2<br />

4<br />

y<br />

y<br />

y<br />

2<br />

1<br />

2<br />

2<br />

2<br />

3<br />

2<br />

4<br />

y<br />

<br />

1<br />

<br />

<br />

b1<br />

<br />

,b =<br />

,j = <br />

1<br />

.<br />

<br />

b2<br />

<br />

1<br />

<br />

<br />

1<br />

<br />

In the general case, it turns out Xb-j admits a unique least squares solution of minimal<br />

norm b = X + j, while the precision of fit can be measured by comparing the image, Xb, to<br />

the desired value j. Thus, the fraction R 2 = ||Xb|| 2 /||j|| 2 is used as a quantity giving an idea<br />

about the accuracy of the proposed approximation. The notation X + represents the socalled<br />

“generalized inverse” in the sense of Moore-Penrose, which corresponds to the<br />

regular inverse if it exists (that is, when X is invertible, and thus, when an exact solution<br />

can be computed). We will not bother here about explaining computational aspects of this<br />

X + when X is not invertible, since mathematical software such as MATHEMATICA TM ,<br />

allows computing this generalization of the notion of the inverse, called “pseudo inverse”,<br />

without effort.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 61

1<br />

<br />

In the given example, X = <br />

1<br />

0<br />

<br />

1<br />

1<br />

<br />

7 <br />

4<br />

and b = X + <br />

j = 11<br />

<br />

2<br />

4<br />

and R 0. 932 .<br />

2<br />

<br />

<br />

1<br />

<br />

11<br />

The set-up can be modified to find, for instance, the n th degree polynomial<br />

y b b x b x<br />

0<br />

1<br />

2<br />

2<br />

b x<br />

3<br />

3<br />

b n x<br />

which best fits a set of given points (x i , y i ). In this case, the expression X.b-j should be<br />

minimal, and the involved matrices are:<br />

1<br />

<br />

X =<br />

1<br />

<br />

<br />

<br />

1<br />

x<br />

x<br />

x<br />

<br />

1<br />

2<br />

m<br />

x<br />

x<br />

x<br />

2<br />

1<br />

2<br />

2<br />

<br />

2<br />

m<br />

<br />

<br />

<br />

<br />

n<br />

x b<br />

1<br />

1 1<br />

n <br />

x2<br />

<br />

<br />

, b = <br />

b2<br />

,j = <br />

1<br />

.<br />

<br />

<br />

n<br />

<br />

xm<br />

<br />

<br />

b n <br />

1<br />

<br />

Again, X.b-j admits a unique “least squares solution of minimal norm” b = X + j while<br />

the precision of fit can be measured by R 2 = ||Xb|| 2 /||j|| 2 .<br />

Application 1: a chain curve approximates Gaudi’s Paelle Guell better than a<br />

parabola<br />

Gaudi is known for his use of hyperbolic cosine functions. He used bags suspended by<br />

ropes which he inverted to get chain curves. However, the question remains whether or not<br />

a non-informed spectator can actually see this in Gaudi’s work, that is, if a spectator can tell<br />

that a Gaudi gate, entrance or window (as for example in the Paelle Guell, fig. 2a) has a<br />

hyperbolic cosine shape. Couldn’t he conclude that the observed curves are, for instance,<br />

parabolas?<br />

n<br />

Fig. 2a<br />

I proposed this question during a course at the Sint-Lucas Institute for Architecture<br />

(Belgium), and student Sil Goossens came up with the following example, for which he<br />

62 DIRK HUYLEBROUCK – Curve Fitting in Architecture

concluded the parabola with equation y = -0.811x² + 5.704x + 2.643 came pretty close, but<br />

not close enough. He claimed “the Gaudi entrance should be considered as a chain curve”.<br />

Fig. 2b. Gaudi’s Paelle Guell gate in Barcelona, some measurements made by a student, and a<br />

parabola fitting well, but not well enough<br />

How can this idea be substantiated? First, checking the coordinates, they look fairly<br />

correct, but we make a change of coordinates to situate the top in (0, 1), since this is a more<br />

straightforward choice when a hyperbolic cosine is expected. In addition, the x-axis is scaled<br />

down by 10% so that all x

Now: y = 107.5-105.8 . Cosh[x], and it fits at 95.3%. Not only do we wonder if this 1%<br />

of difference could be noticed, but further, the result seems to contradict the student’s<br />

conclusion. Looking carefully at the picture, we see the photo is slightly skew with respect<br />

to the observer, and because of this lack of symmetry, the proposed hyperbolic cosine will<br />

never give a nice fit, of course. Thus, we can try to take into account the imperfections in<br />

the picture, and propose a symmetric interpretation, estimating the second coordinates by<br />

modifying the given values slightly:<br />

x13=-x1; x12=-x2; x11=-x3; x10=-x4; x9=-x5; x8=-x6;<br />

j={-7.5, -4.4, -2.55, -.4,.25, .75, 1, .75, .25, -.4, -2.55, -4.4, -7.5};<br />

X={ {1,x1,x1^2,x1^3,x1^4,x1^5,x1^6},…, {…,x13^6}};<br />

This produced the following outcome:<br />

16<br />

2<br />

y 1.019<br />

6.11510<br />

x 20.95x<br />

1.1010<br />

x 60.7x<br />

9.910<br />

x 8.66.96x<br />

13<br />

fitting at 99.875%. If we keep in mind the series for Cosh(x):<br />

we see that:<br />

3<br />

4<br />

13<br />

<br />

2 4 6<br />

2 4 6<br />

x x x x x x<br />

2 Cosh( x)<br />

2 1<br />

... <br />

1<br />

...<br />

2! 4! 6!<br />

<br />

2! 4! 6!<br />

2<br />

5<br />

2<br />

4<br />

6.48x 6.19x 9.3<br />

<br />

2 4 6 x x x<br />

x<br />

y 1<br />

21x<br />

61x<br />

867x<br />

1<br />

21.2! 1454<br />

867.6!<br />

1<br />

<br />

2! 4! 6! 2! 4! 6!<br />

Thus, we should try a different hyperbolic cosine, using a Cosh[a.x] expression, where the<br />

coefficient a is somewhere in the 6.19 or 9.3 range. Now y = 1.34 - 0.36 . Cosh[9.7x] fits at<br />

99.88% and this indeed beats the closest possible parabola, which is y = 1.84 - 52.12x 2 ,<br />

fitting at only 96.75%. This 3% difference in closeness of fit can indeed be noticed, as the<br />

illustration shows.<br />

4<br />

6<br />

6<br />

6<br />

64 DIRK HUYLEBROUCK – Curve Fitting in Architecture

Fig. 3. The catenary and parabola compared to the initial picture. Data were changed putting the<br />

picture in a frontal position: only the left half is to be considered.<br />

Application 2: Gaudi’s Collegio Teresiano can be seen either as a catenary or as a<br />

parabola<br />

The previous result was nevertheless contradicted by another architecture student, Klaas<br />

Vandenberghe, who claimed some of Gaudi’s work shows no hyperbolic cosines, but<br />

parabolas. His examples were the parallel arcs of the Collegio Teresiano. Again, the teacher<br />

was asked to set things straight and to decide who was right.<br />

Fig. 4: The original picture and the obtained catenary and parabola, which<br />

are so close they cannot be distinguished<br />

The procedure is similar: we measure the data as well as possible on the provided<br />

picture, and use the pseudo-inverse algorithm. We start with a 6 th degree polynomial:<br />

y 1.026 3.3310<br />

15<br />

x 5.3x<br />

for which the fit is R 2 = 99.99%. Thus:<br />

2<br />

5.310<br />

14<br />

x<br />

3<br />

29.5x<br />

4<br />

1.8110<br />

13<br />

x<br />

5<br />

100.9x<br />

6<br />

,<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 65

y 1<br />

5.3x<br />

(3.2x)<br />

1<br />

2!<br />

2<br />

2<br />

29.5x<br />

4<br />

(5.1x)<br />

<br />

4!<br />

100.9x<br />

4<br />

6<br />

(6.4x)<br />

<br />

6!<br />

This again reminds us of the hyperbolic cosine series, and<br />

x x<br />

x<br />

1<br />

5.3<br />

2! 29.5<br />

4! 100.96!<br />

...<br />

2! 4! 6!<br />

6<br />

...<br />

y = -0.7468 + 1.75.Cosh[2.8x]<br />

gives a 99.988% fit. A parabola can fit nicely too, as y = 0.985 + 7.63x 2 , shows, with a<br />

99.985% fit. This difference of 0.003% cannot be distinguished with the naked eye.<br />

Application 3: The profile of a nuclear power plant is a hyperbola, or even, an<br />

ellipse<br />

2<br />

4<br />

6<br />

Fig. 5. Determining the profile of a nuclear plant<br />

Consider fig. 5, from which the following values were drawn:<br />

x 2.7; x<br />

1<br />

y 0; y<br />

1<br />

2<br />

2<br />

2.3; x<br />

1; y<br />

3<br />

3<br />

2; y<br />

2; x<br />

4<br />

4<br />

3; y<br />

1.7; x<br />

5<br />

5<br />

4; y<br />

1.47; x<br />

6<br />

5; y<br />

7<br />

6<br />

1,39; x<br />

6.<br />

We compute the general conic section with equation<br />

x 2 b 1 + y 2 b 2 + xb 3 + yb 4 + xyb 5 – 1 = 0<br />

7<br />

1.45;<br />

66 DIRK HUYLEBROUCK – Curve Fitting in Architecture

The minimal norm least squares solution follows from:<br />

X={{x1^2,y1^2,x1,y1,x1*y1},{x2^2,y2^2,x2,y2,x2*y2},{x3^2,y3^2,x3,y3,x3*y3},{x4<br />

^2,y4^2,x4,y4,x4*y4},{x5^2,y5^2,x5,y5,x5*y5},{x6^2,y6^2,x6,y6,x6*y6},{x7^2,y7<br />

^2,x7,y7,x7*y7}};<br />

j={1,1,1,1,1,1,1};<br />

B=PseudoInverse[X].j<br />

Norm[XB]^2/Norm[j]^2<br />

It turns out to be<br />

-0.11x 2 - 0.0105y 2 + 0.67x + 0.19y - 0.06xy = 1 at 99.9996%.<br />

Surprisingly, this is an ellipse. Still, if we lift the x-axis over 5 units, we can propose the<br />

standard form equation for a hyperbola: x 2 b 1 + y 2 b 2 = 1, where b 2 should be negative. Now:<br />

In:<br />

x1=2.7; x2=2.3; x3=2; x4=1.7; x5=1.47; x6=1.39; x7=1.45;<br />

y1=-5; y2=-4; y3=-3; y4=-2; y5=-1; y6=0; y7=1;<br />

X={{x1^2,y1^2},{x2^2,y2^2},{x3^2,y3^2},{x4^2,y4^2},{x5^2,y5^2},{x6^2,y6^2},{x<br />

7^2,y7^2}};<br />

j={1,1,1,1,1,1,1};<br />

PseudoInverse[X].j<br />

Norm[X.PseudoInverse[X].j]^2/Norm[j]^2<br />

Out<br />

{0.508687, -0.108291}<br />

0.998771<br />

This is indeed a hyperbola, which comes as close as 99.8%. Thus, we can faithfully claim<br />

that the shape of a nuclear plant is a hyperbola, and that the adaptations on the top of the<br />

building, for reasons of resistance to wind, can hardly be detected.<br />

Fig. 6. The profile of a nuclear plant, and the approximating shapes, as an ellipse (middle) and a<br />

hyperbola (right)<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 67

Application 4: golden rectangles<br />

Golden section “interpretations” of the Mona Lisa are popular and the influence of The<br />

da Vinci Code surely did not diminish its appeal. Even Wasler’s excellent book [2001]<br />

presents a strange inexplicable shift in the rectangles covering the body, so we’ll concentrate<br />

on the facial proportions, which seem more observable.<br />

Fig. 7. The cover of Walser’s book (left), and a set of data points for the Mona Lisa (right)<br />

Pointing the cursor on the scanned image, we get numbers that have rather useless and<br />

far too many decimals places, but it takes more time and energy to judge about their utility<br />

than to copy them from the computer output:<br />

x1 = 17.496; y1 = 4.871; x2 = 19.129; y2 = 7.143; x3 = 19.200; y3 = 10.551;<br />

x4 = 20.549; y4 = 13.107; x5 = 21.259; y5 = 16.160; x6 = 21.046; y6 = 19.426;<br />

x7 = 20.549; y7 = 22.337; x8 = 20.052; y8 = 25.248; x9 = 18.064; y9 = 27.804;<br />

x10 = 16.573; y10 = 30.644; x11 = 14.372; y11 = 32.490; x12 = 12.313; y12 = 33.413;<br />

x13 = 10.396; y13 = 33.910; x14 = 8.195; y14 = 33.839; x15 = 5.852; y15 = 33.697;<br />

x16 = 3.722; y16 = 33.058; x17 = 1.947; y17 = 31.993; x18 = .740; y18 = 29.792;<br />

x19 = 0.456; y19 = 27.378; x20 = .243; y20 = 24.822; x21 = 0.243; y21 = 12.397;<br />

x22 = 0.811; y22 = 9.912; x23 = 1.450; y23 = 7.427; x24 = 2.657; y24 = 5.013;<br />

x25 = 4.361; y25 = 3.380; x26 = 5.710; y26 = 1.889; x27 = 7.688; y27 = 0.571;<br />

x28 = 10.224; y28 = 0.571; x29 = 13.805; y29 = 1.541;<br />

X = {{x1^2, y1^2, x1, y1, x1*y1}, … {x29^2, y29^2, x29, y29, x29*y29}};<br />

j = {1, … 1};<br />

PseudoInverse[X].j<br />

Norm[X.PseudoInverse[X].j^2/Norm[j])^2<br />

The result is the ellipse<br />

fitting at R 2 = 99.55%.<br />

- 0.0078x 2 - 0.0033y 2 + 0.1686x + 0.1241y - 0.0009xy = 1<br />

Now the minimum corresponds to x=10.8697…, while the maximum corresponds to<br />

8.99537…, and the difference in y-values is 34.014. The (horizontal) width is 20.9757…-<br />

68 DIRK HUYLEBROUCK – Curve Fitting in Architecture

(-1.110…) = 22.09, and the proportion is N[34.014/22.09] = 1.54.... This is closer to, say,<br />

1.5, than to the golden number of 1.618...<br />

Fig. 8. An ellipse that fits to 99.55%, showing that the vertical to horizontal ratio is 1.54<br />

Conclusion<br />

The least-squares method, known from celestial mechanics and since applied in the most<br />

various areas, could well be used for geometric interpretations in art as well, especially since<br />

modern software makes the computational difficulties nonexistent. One can only wonder<br />

why such a widespread and straightforward mathematical technique has not been applied<br />

before. Today, some prefer jumping immediately to modern computer machinery in visual<br />

recognition of patterns, but such mathematical overkill is not necessary and can even turn<br />

artistic minds away from the (beloved!) topic of geometric interpretations in architecture.<br />

References<br />

CAMPBELL, S.L. and C.D. MEYER, Jr. 1979. Generalized Inverses of Linear Transformations.<br />

London: Pitman Publishing Limited.<br />

MARKOWSKY, George. 1992. Misconceptions about the Golden Ratio. The College Mathematics<br />

Journal 23 (January 1992): 2 - 19.<br />

SPINADEL, Vera W. de. 1998. From the Golden Mean to Chaos. Buenos Aires: Nueva Libreria S.R.L.<br />

WALSER, Hans. 2001. The Golden Section. The Mathematical Association of America.<br />

WASSELL, Stephen R. 2002. The Golden Section in the Nexus Network Journal. Nexus Network<br />

Journal 4, 1 (Winter 2002): 5-6.<br />

About the author<br />

Dirk Huylebrouck obtained his Ph.D. in algebra at the University of Ghent. He worked in Africa for<br />

about twelve years, at the Universities of Congo (Bukavu, Kinshasa) and Burundi, interrupted by<br />

assignments at the Portuguese University of Aveiro and at the Overseas Divisions of American<br />

universities in Europe (such as Maryland and Boston University). Since about ten years, he teaches at<br />

the Department for Architecture Sint-Lucas Brussels (Belgium). His research on generalized inverses<br />

was awarded by a quotation in the book "Current Trends in Matrix Theory 1987", and his study of<br />

zèta(3) the “Lester Ford Award 2002” of the Mathematical Association of America. As editor of the<br />

“Mathematical Tourist” column in The Mathematical Intelligencer, he paid attention to historical<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 69

and artistic aspects of mathematics from Tibet over Africa and Europe to Siberia. A mathematical<br />

artifact from Central Africa inspired him to write a book, from which he presented many talks,<br />

publications and TV-collaborations. He currently tours around Belgium in a multicultural<br />

mathematics show together with the percussion trio “Dakar Electric”.<br />

70 DIRK HUYLEBROUCK – Curve Fitting in Architecture

Giulio Magli<br />

Dipartimento di Matematica<br />

Politecnico di Milano<br />

P.le L. da Vinci 32<br />

20133 Milano, Italy<br />

giulio.magli@polimi.it<br />

Keywords: orthogonal town<br />

planning, polygonal walls<br />

Research<br />

Non-Orthogonal Features in the<br />

Planning of Four Ancient Towns of<br />

Central Italy<br />

Abstract. Several ancient towns of central Italy are characterized<br />

by imposing circuits of walls constructed with the so-called<br />

polygonal or “cyclopean” megalithic technique. The date of<br />

foundation of these cities is highly uncertain; indeed, although<br />

they all became Roman colonies in the early Republican<br />

centuries (between the fifth and third centuries B.C.) their first<br />

occupation predates the Roman conquest. It is the aim of the<br />

present paper to show – using four case-studies – that these<br />

towns still show clear traces of an archaic, probably pre-Roman<br />

urbanistic design, which was not based on the orthogonal “rule”,<br />

i.e., the town-planning rule followed by the Greeks, Etruscans<br />

and Romans. Rather, the layouts appear to have been originally<br />

planned on the basis of a triangular, or even star-like, geometry,<br />

which therefore has a center of symmetry and leads to radial,<br />

rather than orthogonal, organization of the urban space.<br />

Interestingly enough, hints – so far unexplained – pointing to<br />

this kind of town planning are present in the works by ancient<br />

writers as important as Plato and Aristophanes, as well as in the<br />

comment to the Æneid by Marius Servius.<br />

1 Introduction<br />

At the end of the Hellenic Dark Age (around the eighth to seventh centuries B.C.) the<br />

Greeks began the expansion which soon led to the foundation of several towns (“colonies”)<br />

in a wide area of the Mediterranean Sea. All such towns have been planned on the basis of<br />

an orthogonal grid, which divides the urban space in equal rectangular blocks. This is true,<br />

for instance, for the oldest colonies (e.g., Selinunte) already in the early sixth century B.C.,<br />

while in the fifth century, with the reconstruction after the Persian War (e.g., of Miletus,<br />

fig. 1) the orthogonal grid became a rule, theorized, at least according to what has been<br />

referred by Aristotle, by the architect Hippodamus [Castagnoli 1971].<br />

The organization of the streets on the basis of orthogonal sectors was developed, more or<br />

less simultaneously with the Greeks, by the Etruscans. We can be certain of this because,<br />

although most of the Etruscan towns were reorganized, or even completely rebuilt by the<br />

Romans (so that their original urbanistic design is uncertain), one Etruscan town was<br />

destroyed by the Celts before the Roman expansion: Misa (today Marzabotto). The<br />

archaeological excavations at Misa have shown that the town was planned on the basis of an<br />

orthogonal grid oriented to the cardinal points within 2° of error [Mansuelli 1965]. Many<br />

of the blocks of the grid were traced on the ground but never edified, showing that the<br />

planners were foreseeing a wide development of the town (fig. 2).<br />

Nexus Network Journal 9 (2007) 71-92 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 71<br />

1590-5896/07/010059-22 DOI 10.1007/S00004-006-0030-X<br />

© Kim Williams Books, Turin

Fig. 1. The layout of Miletus, based on a rigid orthogonal grid<br />

Fig. 2. Plan of Misa (Marzabotto)<br />

72 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

It should, of course, be noted that the Etruscans were in close trading contact with the<br />

Greeks, so that the degree of the cultural interchange in the process which led to<br />

orthogonal town planning is unclear. In any case, Misa shows, in addition to the<br />

orthogonal grid, the orientation of the streets system to the cardinal points, a thing which is<br />

barely visible in Greece. This is a reflection of the complex foundation ritual (to be<br />

discussed in section 2) which, at least according to many Roman historians, was elaborated<br />

by the Etruscans and directly inherited by the Romans, who combined the orthogonal grid<br />

with the inspiring principle of the so-called castrum (military camp) criss-crossed by two<br />

main roads. The structure of the Roman grid was thus based on two main orthogonal axes,<br />

the Cardus, oriented (at least in principle) north-south, and the Decumanus, oriented eastwest,<br />

corresponding to four main gates at their ends [Rykwert 1999]. Layouts based on this<br />

principle can be seen in all the towns of Roman foundation from the middle of the third<br />

century B.C. onward (see for instance the plan of Augusta Praetoria, today Aosta, founded<br />

around 25 B.C., fig. 3).<br />

Fig. 3. Plan of Roman Augusta Praetoria, today Aosta<br />

Sometimes the disposition of the grid followed health criteria, such as those<br />

recommended by Vitruvius in function of the winds; in some other cases it was rather<br />

dictated by symbolic reasons, as shown by the astronomical alignments observed in towns<br />

such as Augusta Bagiennorum [Barale et al. 2002] and Bologna [Incerti 1999].<br />

It is worth mentioning that orthogonal town planning remained the rule up to the<br />

beginning of the twentieth century, and it is still today considered the best method of town<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 73

planning, at least in the cases of heavy car circulation [Southworth and Ben-Joseph 1997].<br />

Throughout the world there have been very few exceptions to this rule in the last 2500<br />

years, a notable one being the radial organization of the human space which was a<br />

fundamental characteristic of the Inca, as shown by the radial sectors in which the capital of<br />

the Inca empire, Cusco, was symbolically divided (see [Magli 2006a], and complete<br />

references therein).<br />

All in all this is, briefly, what can be observed in the layout of Greek and Etruscan<br />

towns, as well as in the plan of all those towns whose Roman foundation is certain.<br />

However, there exist enigmatic and never explained passages of ancient Greek authors as<br />

important as Plato and Aristophanes, as well as a somewhat famous comment to the Æneid<br />

written by Marius Servius, in which these authors seem to refer to a completely different<br />

kind of town planning, which is triangular or even radial; up to the present, no convincing<br />

explanation of such passages is available. As far as the present author is aware, however,<br />

nobody has ever tried to verify if there actually are ancient towns showing the traces of a<br />

planning based on a triangular, rather than orthogonal, symmetry. The aim of the present<br />

paper is to carry out such an analysis, considering as case-studies four among the most<br />

beautifully preserved settlements of central Italy, characterized by imposing circuits of walls<br />

constructed with the so-called polygonal or “cyclopean” megalithic technique.<br />

2 The Etruscan-Roman foundation ritual<br />

Before entering into the characteristics of the layouts of these towns, it is worth making<br />

a brief discussion of the symbolism associated with the town foundation and planning, at<br />

least according to the texts which have survived. Indeed, Roman historians such as Varro,<br />

Plutarch and Pliny the Elder report that the foundation of towns was governed by a ritual<br />

which was directly inherited from the Etruscans and governed by the rules written in the<br />

sacred books called Disciplina. The Disciplina was the collection of writings of the<br />

Etruscan religion, which was thought of as having being revealed to humanity by the gods.<br />

These books are long lost, but accounts on them have survived (for instance the work De<br />

Divinatione by Cicero) so that we know that they were composed of three parts. First of all,<br />

the libri haruspicini, which dealt with divination (the interpretation of God’s will) made by<br />

the priests called Aruspexes by “reading” the flight of the birds and the livers of sacrificed<br />

animals, especially sheep; second, the libri fulgurales, on the interpretation of thunders, and<br />

finally the libri rituales, dedicated to all aspects of life, such as the consecration of temples,<br />

the division of the people into tribes, and the foundation of towns. The latter consisted in<br />

observing the flight of the birds and in tracing the contour of the town by a plough, steps<br />

which everybody will recognize in the worldwide famous legend of the foundation of Rome<br />

as well. A fundamental part of all the rituals of the aruspexes was the individuation of the<br />

auguraculum, a sort of terrestrial image of the heavens (templum) in which the gods were<br />

“ordered” and “oriented”. A key “document” about this complex symbolic structure is the<br />

so-called Piacenza Liver (fig. 4).<br />

The Piacenza Liver is a first century B.C. bronze model of the liver of a sheep, in 1:1<br />

proportions, found in a field near Piacenza in the nineteenth century. The upper surface<br />

presents three protuberances (one of them corresponds to the gallbladder); the external<br />

perimeter is divided into sixteen sectors, while the surface shows six sectors disposed in a<br />

circle, and eighteen further regions; each sector or region contains the name of an Etruscan<br />

deity, with some of them repeated (many of these have been identified with the<br />

74 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

corresponding Roman deities, such as Jupiter or Mars). The lower surface is divided into<br />

two regions, having the name of the Sun and of the Moon respectively [Pallottino 1997].<br />

Fig. 4. The Piacenza Liver<br />

Fig. 5. The inscriptions on the Piacenza Liver (from [Aveni and Romano 1994])<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 75

The owner of the Piacenza Liver was certainly an aruspex, and the object was probably<br />

used to teach divination, and/or as a help to the memory when doing the exam of the<br />

sacrificed animals. It is an extremely important find because its sixteen external divisions<br />

show the structure of the Etruscan cosmos (fig. 5).<br />

The deities in each sectors are ordered from the most important/benign to those of the<br />

underworld; since we know from independent sources that these deities were arranged also<br />

in a “geographical” order starting from the north and moving in what we would call the<br />

clockwise direction, it follows that the Piacenza Liver was also meant to be “oriented” to<br />

the cardinal points [Aveni and Romano 1994, Pallottino 1997]. Once oriented, the liver<br />

itself was to become an image of the cosmos reported on the earth, and, by analogy, so was<br />

the temple, or the city, that the aruspex was ritually founding. The “center” of the city,<br />

called mundus by the Romans, was therefore an icon of the center of the world, and<br />

contained a “deposit of foundation” in which the first produce of the fields and/or samples<br />

of soil from the native place of the founders was buried. Archaeological proofs of the<br />

foundation rituals have been discovered in the excavations of the Etruscan towns Misa<br />

[Mansuelli 1965] and Tarquinia [Bonghi-Jovino 1998, 2000]. From the Roman period<br />

(around the first century B.C.) an example of auguraculum is known from the city of<br />

Bantia [Torelli 1966]. It is composed by nine stone cylinders (cippus) which were disposed<br />

on the ground to identify the eight main divisions of the cosmos (a simplified version of the<br />

sixteen Etruscan divisions) and the center. The center itself was dedicated to the sun, while<br />

the other cylinders carry inscriptions which recall the role of the birds which come from the<br />

corresponding direction; for instance, the north-east one says BIVAV that is B[ENE]<br />

IU[VANTE] A[VE] (bird bringing a good omen) while the north-west cippus has the<br />

inscription CAVAP, that probably means C[ONTRARIA] AV[E] A[UGURIUM]<br />

P[ESTIFERUM] (bird who comes from a bad place, bringing a pestiferous omen). Finally,<br />

very interesting traces of the foundation ritual, to be discussed in detail below, have been<br />

found in the city of Cosa.<br />

The profound relationship between divination, cosmos and foundation ritual is thus<br />

indubitable. However, the radial division of the Cosmos, reported on the Piacenza Liver,<br />

seems to conflict with the orthogonal organization of the Etruscan-Roman town space; as<br />

mentioned in the introduction, there is also a written text, an enigmatic passage of the<br />

Latin writer Servius Marius Honoratus, who describes the town prescribed by the<br />

Disciplina in a way which it is hard to reconcile with the “squared” town. Servius wrote a<br />

important commentary to Virgil’s Æneid in the fourth century A.D. When commenting<br />

on the marvellous verses of the poem in which the hero is urged to admire the construction<br />

of the newly founded city of Carthage 1 (in the reality the city was founded by Phoenicians<br />

at the end of the ninth century B.C.), Servius writes:<br />

The experts of the Etrusca Disciplina state that those founders of towns who<br />

do not plan the layout with three gates, three main streets, and three temples<br />

dedicated to Jupiter, Juno and Minerva, cannot be considered as people who<br />

obey the rules [author’s translation]. 2<br />

Thus, we have here the description of a sort of radial, or at least triangular town, with<br />

three temples dedicated to the three main gods. The dedication to three gods can be easily<br />

explained in the Roman context because Jupiter, Juno and Minerva formed the Triade<br />

Capitolina, to which usually the main temple was dedicated (although it usually was a<br />

single temple with three cells); however, the town’s layout described by Servius, based on<br />

76 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

the number 3, can hardly correspond to a town planned on an orthogonal grid or, even<br />

less, to the Roman Castrum with four main gates and two main streets. As a consequence,<br />

this passage has generated much confusion in the scholars who have tried to interpret it.<br />

For instance, Bloch [1970] noticed that in Misa (fig. 2) the Acropolis is located in the<br />

north, and therefore, one can conceive the idea that, from the Acropolis, the quadripartite<br />

town would have looked as tripartite, as described by Servius; however, this is quite an ad<br />

hoc explanation, which holds, if it does at all, only for a specific case. We shall instead see<br />

that there are towns in Italy – evidently more ancient than Misa – which might have been<br />

originally planned in accordance to the rules recalled by Servius.<br />

3 The layouts of four cyclopean towns in central Italy<br />

In a wide area of west-central Italy, which extends from Tuscany to Campania, there<br />

exist several towns whose walls were built with the so called cyclopean or polygonal<br />

technique. These walls are constructed of enormous (from one up to twenty tons of weight)<br />

stone blocks, cut in polygonal shapes and fitted together without mortar to form a sort of<br />

giant puzzle. Such a spectacular technique makes its first appearance during the Bronze Age<br />

(around 1500-1300 B.C.) among the Myceneans, and indeed the attribute “cyclopean”<br />

comes from Pausania’s description of the walls of Mycenae and Tyrins. Although much less<br />

famous, the Italian cyclopean towns also achieve the same magnificence and impression of<br />

power and pride which characterize the world-famous Mycenaean sites; usually, however,<br />

the dimensions of the Italian towns are much wider (the perimeter can be as long as 3 km)<br />

and the most striking similarity to the Mycenaean buildings is reached in the so-called<br />

Acropoli. These are imposing megalithic buildings, situated in a dominant position with<br />

respect to the landscape, very similar to the Mycenaean’s citadels in dimensions (of the<br />

order of some tens of thousands of square metres), form (polygonal, like that of the blocks),<br />

accesses (they usually have only one main gate and one postern gate on the opposite side),<br />

contents (the interior usually contained a megalithic basement, perhaps a temple or a<br />

palace), and in their relationship with the landscape, being visible from very far; last but not<br />

least, they are nearly identical in construction technique and show astronomical alignments<br />

which can hardly be attributed to the Romans (for more details on the Acropoli of Central<br />

Italy see [Magli 2006b, 2006c] and references therein). In spite of all this, almost all<br />

polygonal walls in Italy, both in the case of the town walls and in the case of the Acropoli,<br />

are currently dated by the archaeologists many centuries after the Myceneans, that is, to the<br />

first centuries of the Roman expansion, between the end of the sixth and the third century<br />

B.C. (see e.g., [Lugli 1957]). However, with few exceptions no stratigraphy is actually<br />

available to date the walls in their own right, and therefore this dating is essentially based<br />

on the fact that all such places make their first appearance into written history through the<br />

works of the Roman historians (for instance, Livius) who mention the “deduction” of a<br />

Roman colony in the same sites. It is, therefore, assumed that the Romans were responsible<br />

for the construction of the walls after the conquest of the towns. Before the expansion of<br />

the Roman control, however, the ethnic scenario in central Italy was extremely<br />

complicated: the Romans were indeed only one among several Latin tribes, and the region<br />

was inhabited by many other people as well, such as the Hernics and the Volsceans, each<br />

one with their own culture, in active cultural and trade exchanges (or war) with the Latins,<br />

the Etruscans and the Mediterranean area. Thus, most, if not all, the settlements pre-date<br />

the Roman period, and the dating of their walls is actually uncertain, leaving us with the<br />

possibility that the original layout of the towns was conceived before the “orthogonal grid<br />

rule”, during the first centuries of the Iron Age (from the ninth century onward).<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 77

In what follows, we shall investigate this possibility, considering, as said, four cities as<br />

case-studies. I hope, however, to extend this analysis in the future to many other towns in<br />

which the “topographical stratigraphy” is more complicated and deserves further<br />

investigation (towns which certainly deserve further attention are, for instance, Segni,<br />

Amelia and Alba Fucens). Our four “laboratories” here will be two Hernic towns which<br />

have been continuously inhabited up to the present, Ferentino and Alatri, and two towns<br />

whose original foundation is not certain (current opinion is that it is of early Roman age),<br />

which were already abandoned in Roman times, Norba and Cosa.<br />

3.1 Ferentino<br />

Ferentino was certainly inhabited since the seventh century B.C., perhaps earlier. Nobody,<br />

however, knows when the town walls were first constructed, although many archaeologists<br />

place them fully in the Roman period [Quilici and Quilici Gigli 1994]. The walls are in<br />

any case virtually intact, and show, superimposed over the megalithic structure, an<br />

elevation made out of squared blocks which is certainly Roman (fourth-third century B.C.)<br />

(fig. 6).<br />

The same can be seen on the Acropolis, which is an imposing building constructed on a<br />

megalithic basement that is fifteen meters high (fig. 7).<br />

The plan of the town can be seen in fig. 8a. It has five main gates. Among them,<br />

however, only three gates, A1-A2-A3, correspond to the original layout (in particular, the<br />

most important gate, denoted by A1, is the famous Porta Sanguinaria, shown in fig. 6),<br />

while gates B1 and B2 were added in late Republican times (end of the second century<br />

B.C.). It is therefore clear that the original town planning – whether it was Roman or not –<br />

was conceived on the basis of a tripartite structure. Connecting the three original gates one<br />

obtains a isosceles triangle (fig. 8b); it can be readily seen that the Acropolis lies in the<br />

center of this triangle.<br />

The city plan was, therefore, based on just one main axis, oriented roughly north-south;<br />

when the Romans, after the conquest, decided to adapt the urban plan to their “squared”<br />

mentality, this axis became the Cardus of the city. They then opened the two new gates and<br />

the street – roughly oriented east-west – which connects them, playing the role of the<br />

Decumanus. The ideal, commercial, and social center of the city then became the point of<br />

intersection between the two main axes, and at this point was built the forum. What<br />

happened is thus very clear: a triangular city with the Acropolis at the ideal center was remodelled<br />

as a “castrum” city with its center in the Forum. Interestingly enough, during the<br />

medieval age the Ferentino cathedral was built on the pre-existing Acropolis. As a<br />

consequence, the symmetry of the Roman castrum was broken again, since the cathedral<br />

was the “center” of the medieval social life. The people thus felt – perhaps unconsciously –<br />

the necessity of “restoring” the original symmetry of the town, and it was probably for this<br />

reason that two new churches were constructed near the two old city gates A2 and A3. The<br />

new churches, together with the main gate of the city, somewhat reconstructed the<br />

triangular symmetry [Montuori 1996].<br />

78 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

Fig. 6. Ferentino: the south-east sector of the polygonal walls with the gate called Porta Sanguinaria<br />

Fig. 7. Ferentino: the south-west bastion of the Acropolis<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 79

3.2 Alatri<br />

Fig. 8. a, above) Plan of Ferentino; b, below) Plan of Ferentino with the original<br />

triangular layout indicated<br />

Among the cyclopean towns in Italy, Alatri is perhaps the most enigmatic. The city was<br />

built around a small hill, and the town was surrounded by megalithic walls, of which many<br />

remains are still visible today. The Acropolis is placed on the hill, and it is a gigantic<br />

construction, a sort of huge “geometric castle” dominating the center of the town; although<br />

generally not well known, it is one of the most impressive megalithic buildings in the entire<br />

world, and the famous German historian Gregorovius (1821-1891) reported that it made<br />

“an impression greater than that made by the Coliseum” (fig. 9). On the top of the hill<br />

there existed a second megalithic structure, perhaps a palace or a temple, whose giant<br />

basement was used in the Middle Age as the foundation for the Cathedral (some of the<br />

blocks still visible on the right end side of the church reach dimensions of the order of 2 x 2<br />

x 1.5 meters and weigh around 12 tons; in spite of this they are perfectly cut and joined<br />

together at several – up to nine – corners) (fig. 10).<br />

80 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

Fig. 9. Alatri: the north-west sector of the megalithic walls of the Acropolis<br />

Fig. 10. Alatri: the point O, the megalithic basement and the mundus<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 81

Fig. 11. Plan of Alatri. The layout is centered at point O, on the Acropolis<br />

It is already well known that Alatri and its Acropolis were planned in accordance with<br />

rigorous mathematical and astronomical references [Capone 1982; Aveni and Capone<br />

1985; Magli 2006b] and, in particular, it has been shown that the original layout of the city<br />

is based on a radial geometry. The center lies on the Acropolis, near the megalithic<br />

basement (in the point indicated by O in fig. 11), which therefore plays the role of ideal<br />

“focus” of the town.<br />

The walls have six main gates (indicated by P1-P6) and three posterns (p1-p3). All the<br />

main gates, excluding P2, are equidistant from O, and therefore lie on a circle centered in<br />

O. The radius of the circle equals three times the value of the segment OH (which is about<br />

92 m long) connecting the center with the north-east corner of the Acropolis and<br />

indicating the summer solstice sunrise. In addition, the town also shows a sort of curious<br />

quadripartite “symmetry”. Indeed, the north-west sector has two main doors and two small<br />

ones, while the north-east sector two main doors. “Symmetrically dividing by two” with<br />

respect to O, the south-west sector has one main door and the south-east sector one main<br />

door and one small door. It is difficult to attribute all this to chance, because the lines<br />

connecting the pairs of doors, p1-p3, p2-P4, P3-P5, all intersect each other in O. Finally,<br />

near this point is visible a narrow and deep cleft in the rocks, which perhaps indicated the<br />

mundus of the city.<br />

82 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

There can be little doubt about the fact that such a complex geometric layout predates<br />

the Roman period. Interestingly enough, archaeological excavations have shown that the<br />

Roman forum of the city was situated where the main square is located today, in the<br />

northern part of the town. Since in Alatri it was simply impossible to re-convert the<br />

urbanistic design to the “squared” conception, because of the presence of the huge hill of<br />

the Acropolis at the very center, the “social center” of the town was translated to the<br />

northern sector, while the southern one remained a residential quarter without centers of<br />

social aggregation, a role which is still preserved today [Ritarossi 1999].<br />

3.3 Norba<br />

Norba lies at the very end of the Lepini mountains, on a steep ridge which looks towards<br />

the sea, some 80 km south of Rome. The area surrounding the city was inhabited at least<br />

from the fourth century B.C., and, as in Ferentino, at Norba there are strong hints pointing<br />

to a peopling of the town itself at least from the eighth-seventh century B.C., although,<br />

again, the town walls are usually attributed to the Romans and dated to the fourth century<br />

B.C. [Quilici and Quilici Gigli 2001].<br />

Fig. 12. Norba: the south-west front of the ramparts.<br />

Fig. 13. Norba, the main gate or Porta Maggiore<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 83

The city is very big, with a perimeter of some 3 km, and the megalithic walls comprise<br />

three small hills, or Acropoli, each one with temples built on the top. The town was<br />

besieged and all its habitants killed during the Mario-Silla war (around 82 B.C.); since then<br />

the site was never again inhabited, so today it is one of the most beautifully preserved<br />

cyclopean towns of Central Italy (figs. 12,13).<br />

The original layout of Norba was clearly inspired by the number 3: there are indeed<br />

three Acropoli (small hills) and, originally, three main gates A1, A2, A3 (fig. 14a). The<br />

contour of the walls was vaguely circular, but on the south-west side the builders strictly<br />

followed very steep cliffs, giving the town quite irregular boundaries. The interior of the<br />

city exhibits a rigid orthogonal grid which, however, at least in my opinion, cannot be<br />

contemporary with the construction of the walls but must be more recent. Indeed, the<br />

spectacular main gate on the east side, Porta Maggiore (fig. 13), does not lead to any of the<br />

east-west axes of the grid. Conversely, the paved Decumanus, which still today crosses the<br />

entire town, leads not to a gate but to the hill (called Acropoli Minore) located to the south<br />

of the main gate; at the other end the Decumanus enters into the city through gate B1,<br />

which in turn was almost certainly added after the initial construction of the walls [Quilici<br />

Gigli 2003]. The whole internal layout of the town is therefore attributable to a reorganization<br />

of the urban space made by the Romans in the second century B.C., and at<br />

this time the east-west axis became the main axis of the city (fig. 14b).<br />

Fig. 14. a, above) Plan of Norba; b, below) Plan of Norba with the east-west axis highlighted<br />

84 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

The view from the Decumanus was (and still is) really stunning: looking out, the eye<br />

goes to the horizon, while, looking in, one is directly in front of the spectacular ramp that<br />

ascends to the two temples on the top of the hill. One of them is parallel to the axis, while<br />

the orthogonal one was conceived to be viewed from the exterior of the town. Interestingly<br />

enough, the guideline of the Decumanus was fixed in accordance to an astronomical<br />

alignment: it is indeed easy to check that it points to the summer solstice sunset.<br />

3.4 Cosa<br />

The city of Cosa lies directly on the sea, on the Ansedonia promontory in southern<br />

Tuscany. The position of the town, high on the promontory, appears to be ideal for control<br />

of the sea behind, which was very important from the commercial point of view, due to the<br />

mines which are present on the coast above, from the Argentario peninsula to the island of<br />

Elba.<br />

At the base of the hill, at less than 3 km from the town existed an ancient port, equipped<br />

with complex artificial structures. In particular, an impressive channel, the Tagliata Etrusca,<br />

still visible today, is carved out of the rocks; eighty meters long, twelve meters high and two<br />

meters wide, it was probably constructed as part of the works required for managing the<br />

port. However, this work is not mentioned in written sources, and nobody really knows for<br />

certain who built it, when, and why. Although a city named Cosa is mentioned as an<br />

Etruscan settlement by several ancient authors, including Virgil, today most archaeologists<br />

believe that the town was entirely a Roman colony, founded from scratch in the first half of<br />

the third century B.C. [Brown 1951, 1980]. The city was abandoned in the early Imperial<br />

age and therefore, like Norba, it appears as it was two thousand years ago.<br />

The walls of Cosa are masterpieces of polygonal masonry (figs. 15,16) and are equipped<br />

(the only example in Italy) with several towers. These, however, were probably added in<br />

later times with respect to the original construction, since there is no joint between the<br />

blocks of the walls and those of the towers.<br />

Fig. 15. Cosa: the north-west gate<br />

Fig. 16. Cosa: the north-east gate<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 85

Fig. 17. a, above) Plan of Cosa; b, below) Plan of Cosa: two solid lines have been drawn to connect<br />

the mundus of the city, in the central cell of the Capitolium, with the two north gates<br />

86 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

Fig. 18. Cosa: plan of the structures under the Capitolium (after [Brown 1980])<br />

The interior of the city was planned on the basis of a rigid orthogonal grid, and this grid<br />

is in accordance with the disposition of the gates, so that – although no internal axis can be<br />

defined Cardus or Decumanus, because none of them connects two gates – the<br />

hippodamean layout might reasonably be considered contemporary with the construction<br />

of the walls (fig. 17a). Curiously enough, however, it seems that the design was inspired<br />

from its very conception by a tripartite division of the urban space. First of all, the city had<br />

only three main gates (further to these, there is only one postern, located near the<br />

Acropolis). Second, during the excavation of the main temple on the Acropolis, the<br />

Capitolium dedicated to the Triade Capitolina, a squared basement, roughly oriented to<br />

the cardinal points, was discovered (fig. 18). The basement is of course more ancient than<br />

the temple above it, and it almost certainly refers to the first phase of construction of the<br />

town. At a few meters behind the basement, on axis with it and at the very center of the<br />

temple’s cells, a natural rocky cleft was found; this probably contained a foundation deposit<br />

of first produce [Brown 1980]. Thus, the archaeologists probably found in this the mundus<br />

of the city, but it does not correspond to a geometrical center of the town’s orthogonal grid,<br />

since the Acropolis lies in a spectacular position at the southern corner of the city walls,<br />

dominating the sea behind. Thus, how should it be interpreted? If we ignore for a moment<br />

the orthogonal grid on which the streets of Cosa were laid out, the mundus actually turns<br />

out to have a geometrical function: the lines connecting it with the two northern doors<br />

divide the city in three quarters which are very similar in size, and, in addition, the line<br />

pointing to the north-west gate is oriented on the meridian (fig. 17b).<br />

4 Discussion and conclusions<br />

We have thus seen that the layouts of four of the ancient towns of central Italy show<br />

urbanistic features based on the number 3 (three gates, three Acropoli, tripartite or even<br />

radial division of urban space) which can hardly be attributed to the Roman period (or, at<br />

least, they do not correspond to what we know about this period). These features appear to<br />

make reference to an older tradition, one perhaps contained in the lost Etruscan books. As a<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 87

matter of fact, the Piacenza Liver is radially divided and exhibits three “hills”, and Servius<br />

states explicitly that the rules for the foundation of cities, which were contained in the<br />

books, were governed the number 3. If this tradition really existed, it would have pre-dated<br />

the period in which the orthogonal grid became the rule, around the early sixth century.<br />

This proposal is independently supported by other data, such as astronomical alignments<br />

(see also [Magli 2006b, 2006c]); and, in view of the new excavations on the Palatino hill,<br />

we know that the first fortification of Rome itself must also be retro-dated from the<br />

standard period – beginning of the sixth century – to the traditional one indicated by the<br />

Roman historians, around the middle of the eighth century B.C. (by the way, due to the<br />

enormous amount of archaeological stratifications, the layout of the archaic town of Rome<br />

remains uncertain, see [Carandini 1997]).<br />

A problem now arises, namely, how are we to understand in which cultural horizon the<br />

“radial” or the “tripartite” geometrical planning should be collocated? It is possible that this<br />

tradition originated in Greece. In fact, as mentioned in the introduction to the present<br />

paper, although no classical Greek city was ever laid out radially, there are some enigmatic<br />

passages by Greek writers that mention a radially planned town [Cahill 2002]. First of all,<br />

one could cite Plato’s famous description of Atlantis in the Critia, in which a circular town<br />

surrounded by concentric water channels is depicted. Secondly, leaving aside the many<br />

controversial questions about this “ideal” city, in his last dialogue, The Laws (written<br />

around 460 B.C.), Plato states the way in which all new cities should be planned by saying:<br />

We will divide the city into twelve portions, first founding temples to<br />

Hestia, to Zeus and to Athena, in a spot which we will call the Acropolis,<br />

and surround it with a circular wall, making the division of the entire city<br />

and country radiate from this point (transl. Benjamin Jowett).<br />

A star-like town also appears in the comedy Birds, written by Aristophanes around 415<br />

B.C. In this comedy, a person called Meton (probably a caricature of the astronomer<br />

Meton of Athens) proposes planning a city in this way:<br />

With the straight ruler I set to work to inscribe a square within this circle; in<br />

its center will be the market-place, into which all the straight streets will<br />

lead, converging to this center like a star, which, although only orbicular,<br />

sends forth its rays in a straight line from all sides (anonymous transl., 1917).<br />

The passage is clearly satiric but, even if Aristophanes was attempting to criticize Meton<br />

(an intention which is far from clear), in any case he depicts an urban plan which is, once<br />

again, star-like. This “theoretical” star-like town has generated much debate, and there is no<br />

satisfactory interpretation available. For instance, the authoritative scholar Francesco<br />

Castagnoli wrote:<br />

lf the comparison is taken literally, the vision of a "PIace de l'Étoile" arises.<br />

But such a plan was not employed until the seventeenth century; it was<br />

totally unknown to the ancient world. Though it is true that the poet can<br />

create before the architect, a less literal interpretation of the passage would be<br />

appropriate: the rays are the four streets which, spreading from the agora,<br />

define the quadripartite city.<br />

Thus, the “rays” should be the four main streets of the squared town, an interpretation<br />

which is at least questionable. However, it is not true that a radial plan was totally<br />

88 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

unknown to the ancient world: it was unknown (or, better, unrealised) in the classical<br />

world, but the radial organization of the inhabited space is present in Greece before the<br />

classical period: it is indeed attested to, for instance, by the Neolithic site of Dimini (fig.<br />

19) and in the Mycenaean citadel of Aghios Georgios near Chalandritsa; further, it should<br />

be noticed that the radial planning of settlements was the rule in Palestine during the Iron<br />

Age [Finkelstein 1988]; see, for example, the layout of the site of Nasbeh, dating around<br />

the ninth century B.C. (fig. 20).<br />

Fig. 19. Plan of the Neolithic settlement of Dimini<br />

Fig. 20. Plan of the Iron Age settlement of Nasbeh, Palestine<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 89

Therefore, it might be that Plato and Aristophanes, like Servius, were referring to old<br />

Mediterranean traditions, filtered during the Hellenic Middle Age, and thus it might well<br />

be that the building techniques and the layouts which are visible in the Italian cyclopean<br />

towns originated in Greece, during the Hellenic Dark Age, or, even before, in the<br />

Mycenaean period.<br />

Note added in proofs:<br />

After the completion of this work, the author became aware of a new, very important<br />

discovery in southern Lazio.<br />

Indeed, archaeologists Lorenzo Quilici and Stefania Quilici Gigli have discovered the<br />

remains of a previously-unnoticed archaic town located on the steep hill called Pianara,<br />

near Fondi. This town, certainly inhabited from the sixth to the fourth centuries B.C., is<br />

probably the one called “Amyclae” by the Roman historians. The town is fortified with a<br />

imposing polygonal wall, and it turns out that the circuit of the walls comprises three hills<br />

and three main gates.<br />

Notes<br />

1. Miratur molem Aeneas, magalia quondam – miratur portas strepitumque et strata<br />

viarum, which roughly means “Look at the size (of the town) Enea, where before was only<br />

rubbish – look at the gates, the street traffic and the industrious people.”<br />

2. Prudentes Etruscae disciplinae aiunt apud conditores Etruscarum urbium non putatas iustas<br />

urbes, in quibus non tres portae essent dedicatae et tot viae et tot templa, Iovis Iunonis<br />

Minervae.<br />

References<br />

AVENI A. and G. CAPONE. 1985. Possible Astronomical Reference in the Urbanistic Design of<br />

Ancient Alatri, Lazio, Italy. Archaeoastronomy 8: 12.<br />

AVENI, A. and G. ROMANO. 1994. Orientation and Etruscan Ritual. Antiquity 68: 545-563.<br />

BARALE, P., M. CODEBO, and H. DE SANTIS. 2002. Augusta Bagiennorum (Regio Ix) Una Città<br />

Astronomicamente Orientata. Turin: Ed. Centro Studi Piemontesi.<br />

BLOCH, R. 1970. Urbanisme et religion chez les etrusques: explication d’un passage fameux de<br />

Servius. In Studi sulla citta` antica : atti del convegno di studi sulla citta` etrusca e italica<br />

preromana. Bologna: Istituto per la storia di Bologna.<br />

BONGHI-JOVINO, M. 1998. Tarquinia: Riflessioni sugli interventi tra metodologia, prassi e problemi<br />

di interpretazione storica. Pp. 41-51 in Archeologia della città. Quindici anni di scavo a Tarquinia,<br />

M. Bonghi Jovino, ed. Milan.<br />

———. 2000. Il complesso sacro-istituzionale di Tarquinia. Pp. 265-270 in Roma, Romolo, Remo e<br />

la fondazione della città, A. Carandini and R. Cappelli, eds. Rome: Electa.<br />

BROWN, F. 1951. Cosa I. History and Topography. Memories of the American Academy in Rome<br />

20.<br />

———. 1980. Cosa: the making of a Roman Town. Ann Arbor: University of Michigan Press.<br />

CAHILL. N. 2002. Household and City Organization at Olynthus. New Haven: Yale University Press.<br />

CAPONE, G. 1982. La progenie hetea. Alatri: Tofani ed.<br />

CARANDINI, A. 1997. La nascita di Roma. Turin: Einaudi.<br />

CASTAGNOLI, F. 1971. Orthogonal town planning in antiquity . Cambridge, MA: MIT press.<br />

FINKELSTEIN, I. 1988. The Archaeology of the Israelite Settlement . Jerusalem: Brill Academic.<br />

INCERTI, M. 1999. The Urban fabric of Bologna: orientation problems. Pp. 3-12 in Atti, VI<br />

international seminar on urban form, R. Corona and G.L. Maffei, eds., Florence: Universita' degli<br />

Studi di Firenze.<br />

LIBERATORE, D. 2004. Alba Fucens: Studi di storia e di topografia. Rome: L’Erma di Bretschneider.<br />

90 GIULIO MAGLI – Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy

LUGLI, G. 1957. La tecnica edilizia romana con particolare riguardo a Roma e al Lazio. Rome: Bardi.<br />

MAGLI, G. 2006a. Mathematics, Astronomy and Sacred Landscape in the Inka Heartland. Nexus<br />

Network Journal 7, 2: 22-32.<br />

———. 2006b. The Acropolis of Alatri: astronomy and architecture. Nexus Network Journal 8, 1: 5-<br />

16.<br />

———. 2006c. Polygonal walls in the Latium Vetus: an archaeo-astronomical approach. In<br />

Proceedings, SEAC 2006 conference, J. Liritzis et al., eds. Mediterranean Archaeology and<br />

Archaeometry 6, in press.<br />

MANSUELLI, G.A. 1965. Contributo allo studio dell’urbanistica di Marzabotto. Parola del passato :<br />

Rivista di studi antichi XX: 314.<br />

MERTENS, J. 1969. Alba Fucens. I-II. Rapports et études. Brussells and Rome.<br />

MONTUORI, F. 1996. Ferentino: Piano di recupero della cinta muraria. In Cinte murarie di antiche<br />

città del Lazio. Comm. Eur.- Progetto Raphael. S. Quirico d'Orcia: Editrice Don Chisciotte.<br />

PALLOTTINO, M. 1997. Etruscologia. Milan: Hoepli.<br />

QUILICI, L. and QUILICI GIGLI, S. 1994. Ricerca topografica a Ferentinum. Pp. 159-254 in Opere di<br />

assetto territoriale ed urbano. Atlante tematico di topografia antica, vol. 3. Rome: L’Erma di<br />

Bretschneider.<br />

———. 2001. Sulle mura di Norba. Pp. 181-244 in Fortificazioni antiche in Italia: età repubblicana.<br />

Atlante tematico di topografia antica, vol. 9. Rome : L’Erma di Bretschneider.<br />

QUILICI GIGLI, S. 2003. Norba: l’acropoli minore e i suoi templi. Pp. 289-321 in Santuari e luoghi di<br />

culto nell’Italia antica. Atlante tematico di topografia antica, vol. 12. Rome: L’Erma di<br />

Bretschneider.<br />

RITAROSSI, M. 1999. Aletrium. Alatri: Tofani editore.<br />

RYKWERT, J. 1999. The Idea of a Town: The Anthropology of Urban Form in Rome, Italy, and The<br />

Ancient World. Cambridge, MA: MIT Press.<br />

SOUTHWORTH M. and E. Ben-Joseph. 1997. Streets and the Shaping of Towns and Cities. New<br />

York: McGraw-Hill.<br />

TORELLI, M. 1966. Un templum augurale d’età repubblicana a Bastia. Rendiconti dell'Accademia<br />

nazionale dei Lincei – Classe Sc. morali storiche filologiche XXI: 293-315.<br />

About the author<br />

Giulio Magli is a full professor of Mathematical Physics at the Faculty of Civil, Environmental and<br />

Land Planning Engineering of the Politecnico of Milan, where he teaches courses on Differential<br />

Equations and Rational Mechanics. He earned a Ph.D. in Mathematics at the University of Milan in<br />

1992 and his research activity developed mainly in the field of General Relativity Theory, with special<br />

attention to problems of relevance in Astrophysics, such as stellar collapse. His research interests<br />

however include History of Astronomy and Archaeoastronomy, with special emphasis on the<br />

relationship between architecture, landscape and the astronomical lore of ancient cultures. On this<br />

subject he recently authored the book Mysteries and Discoveries of Archaeoastronomy, published in<br />

2005 (in Italian) by Newton & Compton.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 91

James Harris<br />

25 West Drive<br />

Plandome, N.Y. 11030 USA<br />

jharris@related.com<br />

Keywords: algorithms, fractals,<br />

rule-based architecture, Frank<br />

Lloyd Wright, Piet Mondrian<br />

Research<br />

Integrated Function Systems and<br />

Organic Architecture from Wright to<br />

Mondrian<br />

Abstract.. The development of an architectural form where the<br />

individual parts reflect the integrated whole has been a design<br />

goal from ancient architecture to the current explorations into<br />

self-organizational structures. Organic architecture, with this<br />

part-to-whole association as an element of its foundation, has<br />

been explored from its incidental use in vernacular structures to<br />

its conscious endorsement by Frank Lloyd Wright. Traditionally<br />

Piet Mondrian has not been associated with organic architecture<br />

but a closer examination of the artistic and philosophical<br />

underpinnings of his work reveals a conceptual connection with<br />

organic architecture.<br />

The development of an architectural form where the individual parts reflect the<br />

integrated whole has been a design goal from ancient architecture to the current<br />

explorations into self-organizational structures. From medieval castles such as the Castel de<br />

Monte, Gothic cathedrals such Reims to Hindu temples [Sala 2000] historically it appears<br />

that our minds are oriented to appreciate buildings constructed with this quality [Salingaros<br />

2001]. Organic architecture, with this part-to-whole association as an element of its<br />

foundation, has been explored from its incidental use in vernacular structures to its<br />

conscious endorsement by Frank Lloyd Wright. Traditionally Piet Mondrian has not been<br />

associated with organic architecture but a closer examination of the artistic and<br />

philosophical underpinnings of his work reveals a conceptual connection with organic<br />

architecture. This relationship is explored through the application of some of nature’s<br />

fundamental structural principles to his artistic style.<br />

The appreciation of beauty is one of the most basic human capacities. Architectural<br />

beauty is developed by the awareness of a balance of order in diversity of the unified whole<br />

and its constituent parts or the “ensemble effect of beauty”[Langhein 2001]. The quest to<br />

understand beauty in architectural design leads us to examine the intrinsic idea of nature<br />

[Bovill 1996].<br />

Gestalt psychology focuses on visual perception to emphasize the dynamic interplay of<br />

parts and whole. Gestalt is usually translated as form or organized structure and is rooted in<br />

German thought of the self-actualizing wholeness of organic forms [Hubert]. It holds that<br />

we have certain tendencies to perceive visual data in organized or configurational terms<br />

[Detrie 2002] and to “constellate” or to see as “belonging together” elements that look<br />

alike, are proximate to each other, are similarly spaced, or are arranged in such a way that<br />

they appear to continue each other. The appearance of parts is determined by and<br />

understood relative to the systematic whole [Behrens; Crowe 1999]. A Gestalt quality is not<br />

concerned with the combination of the various elements per se but in the entity that is<br />

created based on their unity but is still discernible from them [Lyons].<br />

Nexus Network Journal 9 (2007) 93-102 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 93<br />

1590-5896/07/010093-10 DOI 10.1007/S00004-006-0031-9<br />

© Kim Williams Books, Turin

Gestalt theorists believe in an aesthetic dimension of inherent order in nature [Lyons].<br />

Nature has a highly complex and ordered system which we connect to through our<br />

involuntary and subconscious perceptual system as well as our conscious understanding. As<br />

we are a part of nature’s province it is logical that we are structured to appreciate nature’s<br />

underpinnings [Detrie 2002]. Our constructed world is an alternative environment and it is<br />

inherent that there is a coherent connection, subconscious or conscious, between it and the<br />

natural world. In the striving for a harmony between the two worlds it is compelling to<br />

search for the source of nature’s structure.<br />

Jean Piaget, the noted cognitive developmental psychologist, advanced a theory of<br />

structure based on three properties: wholeness, transformation and self regulation.<br />

Wholeness, the “defining mark of structures”, developed from elements that were<br />

subordinated to laws and it is in terms of these laws that the structural whole is defined.<br />

Transformations are laws that govern the structure’s composition and are based on the<br />

overall properties of the whole. Self regulation is the concept that the transformations tend<br />

to develop elements that belong to the structure and preserve its laws [Kranbuehl 2000]. All<br />

of these properties bear the hallmarks of Integrated Function System (IFS) fractals.<br />

In the 1950s Beniot Mandlebrot formalized the study of fractals, which had been<br />

ongoing since the nineteenth century, culminating in his book The Fractal Geometry of<br />

Nature [1977]. Due to the amount of calculations involved, this study was stunted until<br />

the advent of computers. Mandelbrot’s goal was to describe nature with geometry and<br />

numbers in order to illuminate its underlying structure. There are a number of different<br />

types of fractals, including escape time fractals, which are the source of many fractal images.<br />

Fractals based on IFS have been shown to possess nature’s structural traits. Fractals have<br />

been cited as being representative of real plant structures, among other natural structures,<br />

whose distinctive feature is self-similarity based on a recursive procedure called algorithms<br />

for creating these forms. The self-similarity trait manifests itself in the analysis of the fractal<br />

on different scaling levels. An IFS fractal is produced by taking a starting object, such as a<br />

box, as a seed shape or initiator. Copies of the seed shape are manipulated by certain<br />

permitted affine transformations, such as rotation, translation, shearing, and scaling. The<br />

summation of these transformed copies becomes the new seed shape, which is again<br />

transformed by the exact same set of transformations that was used to create it. Each set of<br />

transformations is called an iteration. As the amount of iterations increase the initial seed<br />

shape becomes less evident and the rules used to create the subsequent seed shapes become<br />

more significant. The transformation rules, which are made up of the various affine<br />

transformations, are the essence of the fractal form. The fractal rules are expressed in the<br />

following format, which can be viewed with a text editor (Table 1).<br />

a b c d e f g h i j k l p<br />

0 0 0 0 0 0 0 0 0 0 0 0 0<br />

0 0 0 0 0 0 0 0 0 0 0 0 0<br />

0 0 0 0 0 0 0 0 0 0 0 0 0<br />

0 0 0 0 0 0 0 0 0 0 0 0 0<br />

Table 1.<br />

In this example there would be a set of four transformed copies. The letters a, b, c, d, e,<br />

f, g, h and i are values which describe the rotation, shearing and scaling transformations on<br />

the object in three dimensional space; j, k, and l are the vectors which describe three-<br />

94 JAMES HARRIS – Integrated Function Systems and Organic Architecture from Wright to Mondrian

dimensional translation. P is a probability factor and the summation of that column will<br />

equal 1. The critical determinant elements in an IFS fractal structure are the affine<br />

transformations or rules represented by these values.<br />

A classic IFS fractal is a tree. In fig. 1, the seed shape is shown on the left as a slightly<br />

deformed column. To the right of it is the graphic representation of the transformation<br />

rules which constitute the first iteration. The rules are three-fold: 1) take the seed shape,<br />

reduce it, rotate it slightly to the right, and translate it approximately halfway up the seed<br />

shape; 2) take the seed shape, reduce it, rotate it slightly to the left, and translate it<br />

approximately three quarters up the seed shape; 3) take the seed shape, reduce it, rotate it<br />

slightly to the right, and translate it close to the top of the seed shape. The figure at the<br />

right represents the fractal after three iterations.<br />

Fig. 1<br />

The component parts of an IFS fractal after the first iteration look like a scaled-down<br />

copy of the whole and demonstrates the self-similar character of fractals in general. As<br />

demonstrated in fig. 1 and the table above, IFS fractals contain the “DNA” structure of<br />

natural objects and form the connection between mathematical elements and nature.<br />

Compositions based on IFS fractals can be the source of organic architecture whose<br />

inspiration is the underlying structure of a fractal [Lorenz 2003].<br />

Frank Lloyd Wright believed in the importance of the study of nature as the basis for<br />

establishing an emerging American architecture. In a speech to the Royal Institute of<br />

British Architects he declared, “Modern architecture is a natural architecture—the<br />

architecture of nature, for Nature” [Wright 1939, 10]. Wright’s mentor, Louis Sullivan,<br />

studied botanist Asa Gray and utilized the “manipulation of the organic” in the<br />

development of motifs [Gans and Kuz 2003]. Wright used nature as the basis of his<br />

geometrical abstraction and wanted to extract the geometry he found in nature [Eaton<br />

1998]. He held that “nature means the essential significant life of the thing” and believed<br />

the term organic meant the relationship “in which the part is to the whole as the whole is to<br />

the part and which all devoted to a purpose consistently…some connection with this inner<br />

thing called the law of nature” [Meehan 1987]. These relationships are the essence of IFS<br />

forms and the basis of natural structures.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 95

Fig. 2<br />

One project of Wright’s that has been cited for its fractal qualities is the Palmer House<br />

[Eaton 1998] (fig. 2). The citation of this building points out the misconception that a<br />

repetition of a form, the triangle in this case, constitutes a fractal quality. It is not the<br />

repetition of the form or motif but the manner in which it is repeated or its structure and<br />

nesting characteristics which are important. In fig. 3 I have developed a fractal form which<br />

is similar to the Palmer House plan. In the first iteration, the seed shape is shown in darker<br />

gray and the three transformations are shown in lighter gray. In the second and third<br />

iterations the original seed shape is dropped out and the iterations are color coded in three<br />

shades of gray.<br />

Fig. 3<br />

96 JAMES HARRIS – Integrated Function Systems and Organic Architecture from Wright to Mondrian

Fig. 4. Photograph by Thomas A. Heinz, AIA<br />

© Thomas A. Heinz<br />

Fig. 5<br />

Frank Lloyd Wright only had two high-rises built: the Johnson Wax building in Racine<br />

Wisconsin and the Price Tower in Bartlesville, Oklahoma (fig. 4). In fig. 5 I have<br />

developed an IFS fractal model, utilizing 3D Max Maxscript, which approximates the Price<br />

Tower. The seed shape is an extruded square which represents the shaft of the Price Tower.<br />

The transformation rules consist of scaling, rotations and skewing of the seed shape and can<br />

be categorized into the base, shaft and top portions of the building. The bottom portion<br />

consists of four seed shapes scaled down to small vertical slabs with four additional copies of<br />

the seed shape scaled down to bars. These scaled-down seed shapes are then rotated for a<br />

pinwheel effect in plan. An additional scaled-down seed shape representing the lobby is<br />

skewed and translated to the corner. The middle portion consists of three sets of groupings<br />

stacked vertically. Each grouping consists of a spandrel level, a vision glass level and two sets<br />

of horizontal mullions. Each of these was a vertically scaled-down version of the seed shape.<br />

At the spandrel and vision glass levels four reduced copies of the seed shape are skewed and<br />

rotated as per the pinwheel structure shown in plan and another reduced copy of the seed<br />

shape straddles one corner of the seed shape. The crown portion, similar to the base area,<br />

has four seed shapes scaled down to thin slabs, rotated to produce a pinwheel shape similar<br />

to the base. There is a scaled-down seed shape that remains in the center and another<br />

reduced copy of the seed shape translated over to where it straddles one corner of the seed<br />

shape similar to the shaft portion of the structure. Lastly there are two copies of the seed<br />

shape which are skewed and rotated as one arm of the pinwheel. They are arranged above<br />

each other to continue the vertical arrangement from the middle section.<br />

Wright’s buildings have been reviewed and appreciated by critics for exhibiting “fractal”<br />

qualities at different scales. As one approaches one of his structures there is a progression or<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 97

unfolding of additional elements or details which reflect variations of buildings<br />

characteristics [Lorenz 2003]. Although this experience is not a direct correspondence to<br />

the self-similarity characteristic of IFS based fractals, it is analogous; this is especially<br />

interesting considering that the concept of fractals did not become wide-spread until years<br />

after Wright’s death.<br />

Norman Crowe compared Wright and Le Corbusier, contemporaries of Mondrian, and<br />

the relationship of their architecture to nature [Crowe 1999]. A comparison of their<br />

respective masterpieces, Fallingwater and Villa Savoye, highlights the differences in their<br />

approach. While Villa Savoye represents a “celestial vision” developed out of Greco-Roman<br />

classicism, Fallingwater brings the visitor back to earth with a structure based on an<br />

emerging American architecture enmeshed in its natural setting. Cynthia Schneider has<br />

compared Fallingwater with another example of celestial architecture, the Guggenheim<br />

Museum in New York [Schneider 1999]. The Guggenheim interacts with its environment<br />

in a way similar to the Villa Savoye but, like Fallingwater, it is considered an example of<br />

organic architecture. Thorsten Schnier and John Gero have used principles of genetic<br />

engineering on examples of Wright’s stained glass and Mondrian’s artwork [Schnier and<br />

Gero 1998]. They performed the genetic operations of mutation and cross-over of their<br />

respective genes to create hybrid art forms. These genes are analogous to the IFS codes that<br />

map the transformations of organic fractal architecture.<br />

Wright and Mondrian believed that Truth could be achieved by reducing nature to her<br />

most fundamental forms. They viewed their art as part of the dynamic whole of the cosmos<br />

expressing inner harmonies and the basic truths of the universe. These views are<br />

characteristic of the reduction of nature’s forms to IFS codes and the fractal relation of the<br />

part to the whole.<br />

Mondrian was a disciple of Theosophy and believed religion and art were on parallel<br />

paths: the aim of both was to transcend matter and understand the universal. “Religion<br />

always sought to harmonize man with nature, that is, with untransformed nature”<br />

[Mondrian 1993]. Mondrian revealed his relationship with nature in his essay “Natural<br />

Reality and Abstract Reality,” in which his characters agree that beauty and inspiration are<br />

to be found in nature. The Mondrian character, the Abstract-Real Painter, believes they<br />

need in a sense to look through nature for its transcendental knowledge. He wanted to<br />

pierce the chaotic complexity of natural appearances, to glimpse past this veil to see the<br />

ultimate harmony and “cosmic rhythm” of the universe. The aim of his art was to capture<br />

the underlying structure of nature. Given the enduring popularity of his work “perhaps<br />

Mondrian succeeded in glimpsing through nature’s veil with unmatched clarity” [Taylor<br />

2004]. The relationship of Mondrian’s art to the underlying transcendental structure of<br />

nature makes it an appealing candidate for applications of IFS.<br />

Although Mondrian’s art is associated only with two-dimensional representation, he<br />

frequently wrote about architecture and Neo-Plasticism. In his article “Toward the True<br />

Vision of Reality” Mondrian noted “While Neo-Plasticism now has its own intrinsic value,<br />

as painting and sculpture, it may be considered as a preparation for future architecture”<br />

[Mondrian 1993]. In “Home-Street-City” he wrote, “The application of Neo-Plastic laws is<br />

the path of progress in architecture” [Mondrian 1993] and in “Is Painting Inferior to<br />

Architecture?” he stated, “The new aesthetic for architecture is that of new<br />

painting….Through the unity of the new aesthetic, architecture and painting can together<br />

form one art and can dissolve each other” [Mondrian 1993]. He experimented with<br />

98 JAMES HARRIS – Integrated Function Systems and Organic Architecture from Wright to Mondrian

architectural aspects of his work in some stage designs as well as the design of his own<br />

studio. He was part of the De Stijl movement, which significantly affected architectural<br />

design with some notable buildings.<br />

Mondrian based his art on the presence of scaffolds and the spatial interplay between<br />

their pictorial parts, which is analogous to the manifestation of IFS rules or codes and the<br />

resulting interplay of the parts to the whole. It seems fitting to explore this relationship by<br />

overlaying IFS with Mondrian’s work. In fig. 6, we take the slab seed shape and apply<br />

transformation rules to produce a first iteration of a building facade that is inspired by<br />

Mondrian’s Composition No.1, Composition with Yellow. The upper right element and<br />

the bottom three elements, in addition to being scaled down, are rotated 90° to the right.<br />

The middle element on the left side is scaled down and rotated 180°. The upper left<br />

element is simply scaled down. In the second and third iterations (figs. 7 and 8), they<br />

increasingly exhibit the rhythm Mondrian discussed in his writings and approach the<br />

vitality of his Boogie-Woogie paintings. In figs. 9 and 10 this facade is incorporated with<br />

another Mondrian-influenced IFS facade into an architectural structure.<br />

Fig. 6 Fig. 7 Fig. 8<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO.1,2007 99

Figs. 9, 10<br />

I have been struck with the natural association of IFS and architecture to generate<br />

structures that engender a natural affinity in the observer. In applying IFS to Frank Lloyd<br />

Wright, the principal architect associated with organic architecture, and the artist Piet<br />

Mondrian, who represents the other end of the spectrum, I hope I have demonstrated the<br />

range of possible architectural applications of IFS. I have continued exploring this concept<br />

at http://www.fractalarchitect.com and encourage the reader to continue the experiment.<br />

References<br />

BOVILL, Carl. 1996. Fractal Geometry in Architecture and Design. Basel: Birkhäuser.<br />

CROWE, Norman. 1999. Nature and the Idea of a Man-Made World. Cambridge MA: MIT Press.<br />

DETRIE, Thomas. 2002. Gestalt Principles and Dynamic Symmetry: Nature’s Connection to Our<br />

Built Environment. http://www.public.asu.edu/~detrie/msj.uc_daap/article.html<br />

EATON, Leonard K. 1998. Fractal Geometry in the Late Work of Frank Lloyd Wright. Pp. 23-38 in<br />

Nexus II: Architecture and Mathematics, Kim Williams, ed. Fucecchio (Florence): Edizioni<br />

dell’Erba.<br />

GANS, Deborah and Zebra KUZ, eds. 2003. The Organic Approach to Architecture. London: Wiley-<br />

Academy.<br />

GOLDING, John. 2000. Paths to the Absolute. Princeton: Princeton University Press.<br />

HUBERT, Christian. No date. Gestalt. http://www.christianhubert.com/hypertext/Gestalt.html<br />

KRANBUEHL, Don. 2000. Interplay. Master’s thesis, Virginia Polytechnic Institute and State<br />

University.<br />

http://scholar.lib.vt.edu/theses/available/etd-03152000-14390004/unrestricted/interplay1-<br />

13B.pdf<br />

LANGHEIN, Joachim. 2001. “INTBAU:Proportion and Traditional Architecture. International<br />

Network for Traditional Building, Architecture and Urbanism (INTBAU) 1, 10.<br />

http://www.intbau.org/essay10.htm.<br />

LYONS, Andrew. No date. Gestalt Approaches to the Virtual Gesamtkunstwerk.<br />

http://www.users.bigpond.com/tstex/gestalt.htm.<br />

LORENZ, Wolfgang. 2003. Fractals and Fractal Architecture. Master’s thesis, Vienna University of<br />

Technology. http://www.iemar.tuwien.ac.at/modul23/Fractals/<br />

100 JAMES HARRIS – Integrated Function Systems and Organic Architecture from Wright to Mondrian

MANDELBROT, Benoit. 1977. The Fractal Geometry of Nature. New York: W.H. Freeman.<br />

MEEHAN, Patrick. 1987. Truth Against the World. Wiley-Interscience.<br />

MONDRIAN, Piet. 1993. The Realization of Neo-Plasticism in the Distant Future and in Architecture<br />

Today. In The New Art-The New Life: The Collected Writings of Piet Mondrian. Harry<br />

Holtzman and Martin S. James, eds. Cambridge MA: Da Capo Press.<br />

SALA, Nicoletta. 2000. Fractal Models in Architecture: A Case Study. In Proceedings of the<br />

International Conference on “Mathematics for Living”, Jordan, 18-23 November 2000.<br />

http://math.unipa.it/~grim/Jsalaworkshop.pdf.<br />

SALINGAROS, Nikos A. 2001. Fractals in the New Architecture. Archimagazine.<br />

http://www.math.utsa.edu/sphere/salingar/fractals.html<br />

SCHNEIDER, Cynthia. 1999. Frank Lloyd Wright Lecture. Amsterdam, 18 June 1999.<br />

http://hpi.georgetown.edu/lifesciandsociety/pdfs/franklloydwright061899.pdf.<br />

SCHNIER, Thorsten and John GERO. 1998. From Mondrian to Frank Lloyd Wright: Transforming<br />

Evolving Representations.<br />

http://citeseer.ist.psu.edu/cache/papers/cs/1816/http:zSzzSzwww.arch.usyd.edu.auzSz~thorstenzS<br />

zpublicationszSzacdm98.pdf/from-frank-lloyd-wright.pdf.<br />

TAYLOR, Richard. 2004. Pollock, Mondrian and Nature: Recent Scientific Investigations. Chaos and<br />

Complexity Letters 1, 29.<br />

WRIGHT, Frank Lloyd. 1939. An Organic Architecture: The Architecture of Democracy. Cambridge<br />

MA: MIT Press.<br />

About the author<br />

James Harris has a degree in architecture from Catholic University, an MBA from Fordham<br />

University and is currently a licensed architect in New York City. As a Senior Vice President for the<br />

Related Companies, he has been involved in large scale New York developments, building over four<br />

million square feet of commercial and residential structures in Manhattan.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 101

Charoula<br />

Stathopoulou<br />

University of Thessaly<br />

Papagou 30,<br />

15343 Athens, Greece<br />

stath@rhodes.aegean.gr<br />

Keywords: Cognition, context,<br />

culture, designing activity,<br />

Ethnomathematics, informal<br />

mathematics, teaching of<br />

mathematics<br />

Research<br />

Traditional patterns in Pyrgi of Chios:<br />

Mathematics and Community<br />

Abstract. Ethnomathematical research has revealed interesting<br />

artifacts in several cultures all around the world. Although the<br />

majority of them come from Africa, some interesting ones exist<br />

in Western cultures too. Xysta of Pyrgi are a designing tradition<br />

that concerns the construction of mainly geometrical patterns on<br />

building façades by scratching plaster. The history and the<br />

culture of the community, the way that this tradition is<br />

connected with them, as well as the informal mathematical ideas<br />

that are incorporated in this tradition are some of the issues that<br />

are explored here.<br />

µµ <br />

µµ. <br />

<br />

. <br />

µ <br />

µ, , <br />

. ’ <br />

<br />

, µ <br />

µµ <br />

µµ ’ .<br />

1 Theoretical points<br />

1.1. Introduction<br />

According to D’Ambrosio, “Mathematics is an intellectual instrument created by the<br />

human species to help in resolving situations presented in everyday life and to describe and<br />

explain the real world” [2005, 11]. So, every community depending on its special<br />

environmental and social conditions—not necessarily practical—selects different ways to<br />

answer its own needs.<br />

As Paulus Gerdes notes:<br />

Many peoples do not appear to have referred to the mathematics history<br />

books. This does not mean that these people have not produced<br />

mathematical ideas. It means only that their ideas have not (as yet) been<br />

recognised, understood or analysed by professional mathematicians and<br />

historians of mathematical knowledge. In this respect the role of<br />

Ethnomathematics as a research area resides in contributing with studies that<br />

permit to begin with the recognition of mathematical ideas of these people<br />

and to value their knowledge in diverse ways, including the use of this<br />

knowledge as a starting base in mathematics education [2005].<br />

Nexus Network Journal 9 (2007) 103-118 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 103<br />

1590-5896/07/010103-16 DOI 10.1007/S00004-006-0032-8<br />

© Kim Williams Books, Turin

Ethnomathematical research has shown that all cultures use notions and practices<br />

recognizable as mathematical no matter whether or not mathematics exists as a<br />

distinguishable category of cognition in these cultures. Different cultures present and<br />

develop some common mathematical activities in order to respond to the needs and<br />

requirements of the natural as well as the sociocultural environment. That is to say,<br />

depending on the needs and demands, mathematical activities tend to be developed in<br />

different directions as well as in different degrees in every culture.<br />

According to Bishop [1988] the mathematical activities that are accepted as universal are<br />

counting, measuring, locating, designing, playing, and explaining. These six activities are<br />

adopted as analytical categories by all researchers and very often research focuses on one of<br />

them.<br />

Research regarding the above activities contributes to the acquisition of a deep cognition<br />

about mathematical activities and the ways through which people are educated by them in<br />

every particular culture. Also, it helps us to realise that all cultures have common<br />

characteristics as well as particular ones that distinguish them. Furthermore, the study of<br />

these universal activities results in the recognition and acceptance of each culture’s<br />

contribution to what we today call academic or school mathematics.<br />

1.2. Design activity<br />

The activity of designing concerns “the manufactured objects, artifacts and technology<br />

which all cultures create for their home life, for trade, for adornment, for warfare, for games<br />

and for religious purposes” [Bishop 1988, 39]. An important part of designing concerns the<br />

transformation of some materials, usually from nature, into something that is useful in a<br />

given society with particular conditions.<br />

Design activity exists in every culture. The type of designs depends on the people’s needs<br />

and the available materials. What differs among cultures is what is designed, in what way<br />

and for what purpose. That is to say, in every society depending on its own needs—not<br />

always material—the expression of this particular activity is differentiated.<br />

Some researchers, as for example Pinxten [1983], write about their own impression of<br />

the geometrical and mathematical possibilities of the design forms that appear in several<br />

cultures they have studied. In her book Africa Counts [1973], Zaslavsky presents the richly<br />

geometrical tradition of African societies, part of which is decorative patterns. She also<br />

describes the African architecture that is depicted on houses in the form of elaborate<br />

drawings.<br />

Gay and Cole [1967] note that the Kpelle have developed a technology for the<br />

construction of houses using right angles and circles: “they know that if the opposite sides<br />

of a quadrilateral are of equal length and if the diagonals are also of equal length, the<br />

resulting figure will be a rectangle” [Bishop: 1988, 41]. The Kpelle, although unable to<br />

state this suggestion as a theorem, apply it in their constructions as a culturally acquired<br />

cognition.<br />

Geometrical figures such as the right angle and the orthogonal triangle appear frequently<br />

in all cultures around the world. Circles also play an important role among symbolic<br />

representations, such as in mandalas. Several geometrical figures played important role in<br />

helping people to imagine relations between phenomena.<br />

104 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

Paulus Gerdes [1996, 1999] gives various examples concerning mathematical ideas<br />

incorporated in the design processes of artists in Mozambique as well as in other places in<br />

Africa. Furthermore he emphasizes the necessity of incorporating this cognition into the<br />

curriculum. He maintains that if the hidden mathematics of Mozambique, which he<br />

characterizes as “frozen” mathematics, were “defrosted”, the culture would be revealed and<br />

make it clear that Mozambique’s people, like other people, have produced mathematics.<br />

Among other interesting design traditions connected with mathematical ideas is that of<br />

the quipu of the Incas, studied by Marcia and Robert Asher [1981]. A quipu is an<br />

assemblage of coloured and knotted cotton cords. The colours of the cords, the way they<br />

are connected together, their relative placement, the spaces between them, the types of<br />

knots on the individual cords, and the relative placement of the knots are all part of a<br />

logical-numerical recording. In the tradition of the quipu exist important mathematical<br />

ideas, mostly of graph theory, which the Incas developed much earlier than did the West.<br />

Also, it is important to mention the fact that the representation of the numbers developed<br />

in a way that took into consideration the place value and the representation of zero.<br />

Another interesting expression of design activity is found in the tradition of sona, the<br />

name given by the Tchokwe people of northeast Angola to their standardized drawings in<br />

sand. These were used as mnemonic aids in the narration of proverbs, fables, riddles, etc.<br />

Thus the patterns of sona played an important role in the community’s transmission of<br />

collective memories. The sona patterns depended on the kind of ritual they were used in.<br />

This tradition is also of interest because of the mathematical ideas that are incorporated in<br />

it. Arithmetical relationships, progressions, symmetry, Euler graphs, and the (geometrical)<br />

determination of the greatest common divisor of two natural numbers are some of the<br />

mathematical ideas hidden in sona patterns.<br />

Obviously, each of the design traditions mentioned above is incorporated into its<br />

respective culture and responds to its particular reality.<br />

2 The tradition of Xysta<br />

Xysta (singular xysto ; plural xysta) are a kind of graffiti that appears at the village of<br />

Pyrgi, one of the medieval villages of Chios. Although there are also a few houses in some<br />

other villages with xysta—mostly geometrical patterns constructed by traditional craftsmen<br />

on house façades—those in Pyrgi are considered a particular tradition (fig. 1).<br />

The procedure for their construction is the following. First the craftsman plasters the<br />

façade of the house in one or two layers: the first makes the surface flat, while the second is<br />

the base for xysta. 1 While the material is fresh a layer of whitewash is added. The craftsmen<br />

subdivides the wet surface into zones, and in every zone appropriate patterns are designed.<br />

The pattern is then scratched with a fork into the whitewashed surfaces. The patterns that<br />

appear are the result of the contrast between the scratched whitewash and the plaster (fig.<br />

2).<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 105

Fig. 1<br />

Fig. 2<br />

The main materials that are used for this procedure—depending on the time period—<br />

are different kinds of sand, mortar, whitewash and cement. The instruments that the<br />

traditional craftsmen use are only a lath, dividers with two points, and a fork. The lath<br />

serves two purposes: for the separation of the wall’s surface in zones and for the<br />

construction of straight lines. The dividers are used for the construction of circular figures,<br />

while the fork is used for scratching some areas of the figures in a way the one area is dark<br />

(the scratched one) and the next white and so on.<br />

As will be discussed below, this tradition is very important for the inhabitants’<br />

community and sense of identity. The fact that this is both a cultural practice as well as the<br />

application of interesting mathematical ideas in a traditional art form make xysta an<br />

interesting example of Ethnomathematics.<br />

In this paper the following questions are discussed:<br />

<br />

<br />

<br />

<br />

How is the cultural context connected with this design tradition?<br />

What are the main mathematical ideas that we can see in these patterns?<br />

How is the construction of these patterns a result of informal cognition that<br />

craftsmen acquire through partnership?<br />

How could this be used for teaching some mathematical notions or practices?<br />

106 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

3 Methodological issues: the method of the research<br />

As Ethnomathematics lies in the confluence of mathematics and social (cultural)<br />

anthropology, the main methodology adopted comes from anthropology, namely<br />

ethnography. A commonplace of the researchers who explore cultural parameters is that<br />

“the place of emergence of cultural cognition is ethnography”. It is argued that<br />

ethnographical research constitutes a particular characteristic of modern anthropology that<br />

differentiates it from the other social disciplines [Madianou 1999, 215].<br />

A basic element in ethnography is research on site, with the main characteristic being<br />

participant observation. Participant observation combines participation in peoples’ lives<br />

with a scientific distance that allows the precise observation and reporting of data. Also,<br />

participant observation is a kind of baptism in a culture.<br />

In the framework of an ethnographic work, the researcher remains in the field as long as<br />

necessary in order to acquire access to aspects of life that could not otherwise be easily be<br />

approached in order to select data. In this type of research data can appear a posteriori as<br />

the result of meanings that are attributed in particular contexts and which researcher can<br />

see and interpret after he has been incorporated into the indigenous culture.<br />

Participant observation is considered by some researchers as a method and by others as a<br />

research strategy or technique. Independently of the way we define participant observation,<br />

the majority of researchers consider it the most important as well as the most laborious<br />

method of anthropological research [Madianou 1999, 242]. The reason is that participation<br />

requires the involvement of the anthropologist in everyday activities and community life.<br />

Furthermore, communication through the local language is required. In fieldwork he has to<br />

observe and analyze the incidents in light of their everyday cultural relevance.<br />

The procedure of interviewing is another important element. Open interviews are the<br />

most common type. Although they seem casual, because they have an implicit agenda—in<br />

comparison with the structured interview with its explicit agenda—these kind of interviews<br />

are useful for ethnographic research because they help the researcher understand the way<br />

people think and to compare the opinions of different people.<br />

Another important aspect regarding fieldwork is entrance in the community. Since the<br />

ethnographer doesn’t usually come from the community that he studies, how he or she<br />

approaches the members of the community is a significant issue.<br />

The method that was selected for the present research was one with ethnographical<br />

characteristics. Although the time of my residence during the fieldwork was less than the<br />

usual, the tools of the research were very close to those of ethnography. Residence in the<br />

field, participant observation, and the informal interviews are some of the elements that<br />

determine the research presented here.<br />

4 The fieldwork<br />

Having visited Pyrgi a few times between 1993 and 2005, I had the sense that the<br />

geometrical patterns that the inhabitants create on the façades of their houses were of<br />

mathematical interest. The summer of 2005 I decided to stay in Pyrgi in order to study not<br />

only the kind of patterns but also the reasons why this tradition began and developed here.<br />

Thus most of the material in the present paper was the product of research on site.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 107

4.1 The place and the people: past and present<br />

Pyrgi, also known as the “painted village,” is located in the north of Chios, one of the<br />

Aegean islands. Chios is well-known as the native land of the epic poet Homer. Today it is<br />

famous thanks to mastic, a product of the mastic tree. The inhabitants of Chios, especially<br />

those who come from the south, feel proud of their place because of the fact that while<br />

these trees also exist in other places around the world they don’t produce mastic. Pyrgi is a<br />

mastic village.<br />

Furthermore, Pyrgi is one of the medieval villages of Chios. What differentiates it from<br />

other medieval villages is the fact that it is substantially the same as it was six to seven<br />

centuries ago. Although there is an expansion of buildings, the main part of the settlement<br />

continues to be the same as it was in the past.<br />

There is no sure information about the exact date of the settlement’s construction.<br />

Among other writers, Konstantinos Sgouros [1937] asserts that the village existed before<br />

the possession of Genoa (1346-1566). Another historian, George Zolotas [1928], also<br />

believes that the main core of Pyrgi existed before the possession of Genoa. He also<br />

maintains that the inhabitants of Pyrgi and the nearby settlements were unified for safety<br />

purposes.<br />

The architect-researcher Maria Xyda reports that the conquerors unified the settlements<br />

in order to fortify and organize the ex-Byzantine settlements that produced mastic into a<br />

single settlement [2000, 37]. Xyda estimates that the design of the village happened in<br />

another place. She notes that buildings such as churches were not included in the original<br />

design of the village, and thus claims that the design happened at Genoa. To support this<br />

argument she notes that the medieval villages of Chios were designed in the same way as<br />

Liguria’s villages. The similarities between Chios’s medieval villages and those of Liguria<br />

concern not only the urban layout but the constructive and morphological details of the<br />

houses as well as the use of similar stones [Xyda 2000, 38].<br />

The German sojourner Hohann Michael Wansleben, who visited Chios in 1674, noted<br />

“Pyrgi is very well fortified and it has been built in the Italian way”.<br />

The shape of the settlement originally was a quadrangle. A small tower was built on each<br />

of the four apexes. The houses had neither windows nor doors on the external side, so they<br />

had a view only of the internal side of the settlement. The way those houses were built<br />

formed a wall around the settlement [Proiou 1992, 48]. Two main gates, in the north and<br />

the south, permitted access to the settlement. The houses were arranged like rings. At the<br />

boundaries of every ring the streets were made. These rings were linked by arcs [Xyda 2000,<br />

41]. This form of the village was maintained until the beginning of the twentieth century,<br />

when it started to expand. As a result, today the boundaries are not distinguishable.<br />

The houses of the old part of the village are very similar as far as the design and the<br />

material of construction are concerned. Usually the houses are constituted of three or four<br />

floors.<br />

The type and the arrangement of the place dictate corresponding practises. First of all,<br />

because the inhabitants by definition live very close to each other, they have direct everyday<br />

contact with their neighbours, voluntary or not. What is of great importance for the present<br />

research is the fact that since the houses were narrow and dark, the inhabitants had to<br />

108 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

spend a lot of time outside. In other words, the whole social life of the population happens<br />

in the central square as well as in the streets around the square. The women meet their<br />

friends and their neighbours and do the household duties outside their homes, for example<br />

on the sidewalks in front of their houses, because there are no yards. At the time I was<br />

there I saw women preparing fresh beans, threading tomatoes—in order to dry them—and<br />

undertaking any other kind of activity they could do outside (figs 3-6).<br />

Fig. 3-6. Social life in Pyrgi<br />

The fact that they spent so much time outside their houses doing their everyday duties<br />

seems to have affected the way they realized the exteriors of them. The inhabitants initially<br />

constructed their houses with plain stones. As over time they improved their financial<br />

situation, they were able to plaster their houses in order to protect the walls from the<br />

weather. Later they started to add patterns to the facades of their houses for decoration<br />

purposes. So the practice of plastering, which was initially used simply for protection<br />

against the elements, developed into a way of decorating.<br />

According to some sources, the patterns that they are used are derived from the carpets<br />

that Genovese people—the conquerors—used to put on the outsides of their houses for<br />

decorative purposes. After the Genovese left Pyrgi, the practice of decorating the façades<br />

with carpets was replaced by decorating façades with the patterns on plaster. Others<br />

consider the patterns of Capodocia in Turkey to be the inspiration (see [Xyda 2000]). The<br />

geographical location of Chios, and the Aegean islands in general, appears to explain the<br />

influences of both East and West.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 109

4.2 The evolution of the style of xysta<br />

As already mentioned, there is no sure information concerning either the date or the<br />

origin of the tradition of xysta. It is supposed that a constructive technique developed into a<br />

decorative one. When I asked Maria Xyda if there was an era at which the xysta were<br />

connected with a high status, she answered that the people who had the ability to plaster<br />

their houses and create xysta, in its earliest stages, were the wealthier people of the<br />

community. 2<br />

Although little is known about the date this tradition began, the techniques used from<br />

the beginnings up to the present is well known. Maria Xyda has classified the technique,<br />

the material, the patterns and the style, in general, into five categories [2000: 64-68].<br />

<br />

Xysta of the first period. At this time the patterns were only geometrical<br />

themes—limited types—and the plaster was made of river sand, lime and<br />

straw. The size of the patterns was similar to the size of the stones used as<br />

structural units (fig. 7).<br />

Fig. 7<br />

<br />

Xysta with influences from the East. The xysta of this period were influenced<br />

from the Near East so there was a rich diversity in patterns. Another<br />

characteristic of this period was the tassels they had at the bottom, which<br />

represented carpets. Although the patterns are different from the first period<br />

the material used was the same (fig. 8).<br />

Fig. 8<br />

110 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

Xysta of 1930-1940. The xyst at this time reflect all the previous influences.<br />

The material is not always the same since sea sand and cement were added (fig.<br />

9).<br />

<br />

<br />

Fig. 9 Fig. 10<br />

Xysta after the Second World War. The patterns become increasingly<br />

complicated, while at the same time they abandoned the use of colours. The<br />

material is the same except for the sand, which is now only from the sea, which<br />

is very close to the village (fig. 10).<br />

Contemporary xysta. The patterns are black and white. They aren’t organized<br />

in units, but are more complicated and, in contrast with the previous periods,<br />

they extend over the entire surface of the facades. Furthermore the patterns are<br />

borrowed from some other traditions of craftsmanship, such as carpentry and<br />

blacksmithing. The patterns of this period cover the entire façade without<br />

taking into consideration the doors and windows. The materials that are now<br />

used are cement, a different type of sand that they can buy from the market,<br />

lime, and cinder (fig. 11).<br />

4.3 The entrance in the community<br />

Fig. 11<br />

After visiting Pyrgi a couple of times as a tourist, I decided that studying xysta could be<br />

of great interest in the context of the connections between culture and mathematics. This<br />

special tradition of xysta was important not only from the side of the construction and<br />

designing but also from the side of culture and mathematics. The majority of the patterns<br />

were geometrical constructions that were made by two simple instruments.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 111

The first day I visited Pyrgi as a researcher, in the summer of 2005, I attempted my<br />

entrance in the community through a café located in the central square of the village in<br />

order to meet members of the community (cafeneio = Greek café). As mentioned above<br />

the social life of people takes place in the centre of the village. After this informal discussion<br />

with the inhabitants my informer Elias led me to observe some interesting patterns and also<br />

facilitated my contact with one of traditional craftsmen.<br />

4.4 Material and data selection<br />

The interviews with the inhabitants were informal and semi-structured. Through them I<br />

attempted mostly to understand elements about their identity as inhabitants of Pyrgi and<br />

the connection with the xysta. In contrast, the interviews with the craftsmen and the<br />

architects were more structured because more concrete answers concerning xysta were<br />

expected.<br />

At Pyrgi, I had the chance to meet some very kind and helpful inhabitants who did their<br />

best to facilitate my research. Since the community is a small one, in a very short time,<br />

everybody had been informed that someone was interested in xysta. As a result, while I was<br />

walking down the streets or taking photos some inhabitants approached me and gave me<br />

any information about xysta.<br />

In these discussions I heard several versions of the story of their origin as well as the date<br />

or period when this tradition started. Some of them consider the tradition to have come<br />

from the East (Turkey) and some others from the West (Italy). The location of Chios and,<br />

more generally, the area of Aegean Sea (indeed, the whole of Greece), allowed it to be<br />

influenced by both East and West.<br />

Written documents selected from the local library supplemented the material of the<br />

research on site. In the library I found material concerning the tradition of xysta as well as<br />

the place and the people. Another very important resource for my research was my personal<br />

communication with the architect Maria Xyda. She comes from the island Chios and had<br />

conducted research about xysta in the framework of a European project. This resulted in<br />

her book, The Xysta at Pygri of Chios. In addition to personal communication, this book<br />

was of special interest for my research.<br />

The ethnographical equipment used in the fieldwork were a camera to take some photos<br />

of the many designs of xysta, and paper and pencil in order to take notes during the<br />

fieldwork and to try some original analysis and thoughts. I used a tape recorder for the<br />

interviews.<br />

4.5 Identity of Pyrgi’s inhabitants and xysta<br />

The majority of the inhabitants maintain that the main elements that distinguish their<br />

community from the ‘others’—in the island and generally—are the xysta and their<br />

traditional dance, called pyrgousiko. Some added the traditional clothing, pyrgoysiki, as a<br />

distinguishing factor. The connection of the traditional clothing with the other two cultural<br />

peculiarities is noteworthy: according to their explanations, the designs on the sleeves of<br />

this clothing come from the xysta, although, as I noticed in the folklore museum, only a<br />

part of them had patterns similar to xysta. In any case, the fact that they speak this way,<br />

connecting these traditional elements and their identity, is itself of some importance.<br />

112 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

Figs. 12-14<br />

The following discussion with an 80-year old man is characteristic of the importance of<br />

xysta for the members of community:<br />

“Why do you like to have xysta at your house?”<br />

“Because I’m Pyrgouis (=habitant of Pyrgi). Jesus Christ was born in the<br />

manger and the manger is what he remembers.”<br />

Many others answered the question about their interest in xysta in a similar way, saying<br />

that they like xysta because “these are our tradition”. In some other cases it was tourist<br />

purposes that were emphasized: “The xysta is a means of promotion for Pyrgi, the place is<br />

famous because of its xysta”.<br />

The sign and the design on the t-shirt in figs. 12 and 13 are indicative of the connection<br />

of xysta with what is expected from tourism. Also, it was observed that some modern<br />

buildings, such as hotels, were decorated with xysta. In fig. 14 the interior of a modern<br />

hotel is shown. This hotel was located in the village closest to Pyrgi, which is its seaport.<br />

The majority of this village’s habitants—including the owner of the hotel—came from<br />

Pyrgi. By using the xysta in an alternative way to decorate part of the hotel’s interior he<br />

was declaring the continuity of the tradition.<br />

5 Patterns and mathematical ideas<br />

When studying the tradition of xysta in the framework of culture and mathematics, it is<br />

of great importance to understand how this tradition was incorporated and developed in<br />

this particular culture and what the meaning of it for the community is. On the other side,<br />

it is important to explore the mathematical ideas that are incorporated in them, noting that<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 113

it is about informal mathematics, as xysta are products of craftsmen who have acquired this<br />

cognition through experience, without have been taught something in school.<br />

A few indicative patterns presented here are going to be examined in order to help us to<br />

pick out interesting mathematical ideas. A main mathematical notion that is apparent in<br />

them is the construction of geometrical shapes such as rectilinear or circular figures.<br />

Among others, the notion of transformation is one that appears very often in these<br />

patterns. The kind of transformation used in xysta patterns is isometry, since in all cases the<br />

shapes that are transformed do not change the distances between the points. The isometric<br />

objects are congruent; that is to say, we can turn one into the other just by sliding and<br />

flipping. The kinds of isometries that are found are translation, rotation, and reflection.<br />

More analytically, in every picture we can observe the following mathematical ideas.<br />

Figs. 15-16<br />

The photos in figs. 15 and 16 show different parts of the façade of a church. Before<br />

starting the discussion about the geometrical figures it is interesting to notice some other<br />

elements on these pictures. For example, in the same place we can see religious symbols<br />

(such the cross) and other symbols (such as the half-moon and the point that refers to the<br />

East). Contradictions like this are characteristic of the Aegean islands, because they<br />

combine the culture of the West with that of the East, in particular the Greek Orthodox.<br />

Observing the picture that covers larger surface we can see some zones that separate<br />

different motifs. The subdivision of the surface in parts could be considered as a set that is<br />

separated in sub-sets, a fact that also indicates a mathematical notion.<br />

On the first clear zone we can see triangles that seem to be produced by a translation.<br />

Every triangle (black or white) is the union of two orthogonal triangles, while these<br />

orthogonal triangles come from the division of the rectangle in two equal parts by the<br />

diagonal. For this construction the informal notions that are implicitly used are the<br />

construction of the rectangle; the tracing of the diagonal; the fact that the diagonal divides<br />

the rectangle in two equal parts, and that alternating black and white triangles are<br />

congruent because they are derived from equal rectangles.<br />

In the next zone the main notion is that of symmetry. In this there are two motives that<br />

are repeated. So, first of all we could speak about translation. Furthermore, in every motif<br />

there are two axes of symmetry: the one horizontal, the other vertical. Also the construction<br />

of circles (circular sectors), rectangles, and interstices between lines and circles are<br />

important mathematical ideas.<br />

114 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

The other zones continue with similar mathematical notions and produce variants of the<br />

figures that have already been discussed above.<br />

While the majority of the motives are constituted of rectilinear figures in some cases they<br />

are made of circular figures (fig. 17). As Maria Xyda noted, when the craftsmen didn’t have<br />

enough space to develop a rectilinear figure, as for example in the space between a door and<br />

a window or under a balcony, they constructed circular motives [2000, 63]. The size of the<br />

circles, which they call “moons”, depends on the available area. The only tool that is needed<br />

for the construction is dividers.<br />

Fig. 17<br />

In the preliminary stage for this pattern the craftsman constructed a quadrangle. After he<br />

had determined the centre of this, he traced the three concentric circles. Then, using a<br />

random point of the perimeter of the original circle as the centre, he traces a new circle<br />

whose radius is equal to that of the origin circle (all lines outside the original circle are later<br />

erased). He continued the procedure by tracing new circles, each time using the<br />

intersection of the previous circle with the original one as the new centre. After<br />

constructing the first six parts, which they call “daisy petals”, by finding approximately the<br />

middle of one of the six arcs in which the circles have been divided, he continued with the<br />

same procedure. Scratching the lime of the common area of the petals as well as the area<br />

external them brings into evidence the final design. By continuing to scratch in the middle<br />

circle, the two rings are created. He finished the construction with the semi-circles on the<br />

outside of the circle.<br />

In this pattern a lot of important implicit mathematical notions are present. First of all,<br />

there is the tracing of circles. For the construction of the main circle the centre of the<br />

quadrangle has to be determined. Behind the construction of equal arcs on the main circle<br />

is the equality between the meter of the arcs and the corresponding epicentre angle.<br />

Furthermore, in the main motif twelve axes of symmetry are noticed: six of them are<br />

diameters that connect two opposite points that are the intersection of the circles with the<br />

original circle and the other six are diameters than connect the middle of the arcs that are<br />

opposite.<br />

An interesting façade is shown in fig. 18. It shows a xysto from the period 1930-1940<br />

and it is reproduced from Maria’s Xyda book [2000, 66]. As she notices, xysta of this<br />

period are the most naïf and characteristic since they combine the origins of the Pyrgi<br />

tradition and at the same time give solutions and perspectives for popular art.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 115

Fig. 18<br />

The patterns here are also a combination of geometrical figures. The constructions<br />

concern circles, semi-circles, quadrangles, equilateral and orthogonal triangles (half of a<br />

quadrangle), and the tracing of diagonals.<br />

In the first clear zone, starting from the top the main figures are semi-circles. In this<br />

zone we can talk about translation as well as axial symmetry. In every motif two axes of<br />

symmetry are noticed: one horizontal and one vertical. Similarly in the next zone<br />

translation and axial symmetry are observed. The difference here is that there is only<br />

vertical axis in every motif. The equilateral and orthogonal triangles were produced by<br />

diagonals of the quadrangles.<br />

In the next zone there is a more complicated design. The original figures are rectangles<br />

in which a horizontal line (parallel to the horizontal sides) and diagonals are traced. By<br />

scratching the triangles that are 1/4 or 1/8 of any rectangle we obtain these designs. By<br />

taking one rectangle as one unit, the next rectangle is produced by rotation. In the case that<br />

two adjacent rectangles are considered as one unit, we can talk about a translation.<br />

It should be generally noted that all these patterns are constructed only with two tools:<br />

dividers and a straightedge without markings. Thus these constructions recall the only<br />

constructions that were acceptable in the mathematics of ancient Greece.<br />

6 Some concluding notes<br />

Xysta of Pyrgi is an interesting design activity because of both the significance for its<br />

inhabitants’ culture and the mathematical ideas with which it is connected.<br />

Current approaches of didactics of mathematics discuss the use of everyday<br />

mathematical cognition as well as examples of several cultures in teaching mathematical<br />

notions in the classroom. Patterns like these could be used in the introduction of<br />

mathematical notions such as transformation and symmetries.<br />

By teaching mathematics through patterns, students can not only learn mathematics but<br />

can also understand that mathematics is a component of everyday life. Furthermore, they<br />

are motivated to find information about the corresponding community in which<br />

mathematical ideas are met in traditional activities, and thus see the connection between<br />

culture, cognition, and context.<br />

116 CHAROULA STATHOPOULOU – Traditional Patterns in Pyrgi of Chios: Mathematics and Community

Acknowledgments<br />

The author thanks Chian Architect Maria Zyda for generous help during the fieldwork and<br />

to Professor Ubiratan D’Ambrosio for his remarks.<br />

Notes<br />

1. The number of layers of plaster on the façade depends on the technique as well as on the<br />

material that was used in the construction. The original xysta were only on stone houses,<br />

but now the majority of houses are brick.<br />

2. Personal communication.<br />

References<br />

ASHER, M. and R. ASHER. 1981. Code of the quipu. Grand Rapids: University of Michigan Press.<br />

BISHOP, A.J. 2002. Mathematical Enculturation: A cultural Perspective on Mathematics Education.<br />

Dordrecht, The Netherlands: Kluwer.<br />

D’ AMBROSIO, U. 2005. Preface. Pp. 10-17 in Ethnomathematics: exploring the cultural dimension<br />

of mathematics and of mathematics education, C. Stathopoulou. Athens: Atrapos.<br />

GAY, J. and M. COLE. 1967. The New Mathematics in an Old Culture. New York: Holt, Rienhart<br />

and Winston.<br />

GERDES, P. 1996. Ethnomathematics and Mathematics Education. Pp. 909-943 in International<br />

Handbook of Mathematics Education, J. Bishop et al. (eds). Dordrecht, The Netherlands: Kluwer<br />

Academic.<br />

———. 1999. Geometry from Africa: Mathematical and Educational Explorations. Washington DC:<br />

The Mathematical Association of America.<br />

———. 2005. Geometrical aspects of Bora basketry in the Peruvian Amazon. Mozambique:<br />

Mozambican Ethnomathematics Research Centre.<br />

MADIANOU, D. 1999. Culture and Ethnography: from the ethnographical realism to cultural<br />

criticism. Athens: Greek Letters.<br />

PINXTEN, R. 1983. Anthropology in the Mathematics Classroom? Pp 85-97 in Cultural Perspectives<br />

on the Mathematics Classroom, S. Lerman, ed. Dordrecht, The Netherlands: Kluwer Academic.<br />

PROIOU, I. 1992. The history of Pyrgi. In An Heirloom: Pyrgi of Chios. Athens.<br />

ZASLAVSKY, C. 1973. Africa Counts. Boston: Prindle, Weber and Schmidt.<br />

SGOUROU, K. 1937. History of Chios Island. Athens.<br />

ZOLOTAS, G. 1928. History of Chios. Athens.<br />

ZASLAVSKY, C. 1994, “Africa Counts” And Ethnomathematics. For the Learning of Mathematics 14,<br />

2: 3-7.<br />

XYDA, M. 2000. The xysta of Pyrgi. Chios: Alfa pi.<br />

About the author<br />

Charoula Stathopoulou teaches Mathematics and Didactics of Mathematics at the Special Education<br />

Department of University of Thessaly. Both Ethnomathematics and Ethnomathematics in<br />

conjunction with Mathematics Education are of special interest for her.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 117

Rachel Fletcher<br />

113 Division St.<br />

Great Barrington, MA 01230<br />

USA<br />

rfletch@bcn.net<br />

Keywords: Squaring the circle,<br />

descriptive geometry, Leonardo<br />

da Vinci, incommensurate<br />

values<br />

I Introduction<br />

Geometer’s Angle<br />

Squaring the Circle:<br />

Marriage of Heaven and Earth<br />

Abstract. It is impossible to construct circles and squares of equal<br />

areas or perimeters precisely, for circles are measured by the<br />

incommensurable value pi () and squares by rational whole<br />

numbers. But from early times, geometers have attempted to<br />

reconcile these two orders of geometry. “Squaring the circle” can<br />

represent the union of opposing eternal and finite qualities,<br />

symbolizing the fusion of matter and spirit and the marriage of<br />

heaven and earth. In this column, we consider various methods<br />

for squaring the circle and related geometric constructions.<br />

From the domed Pantheon of ancient Rome, if not before, architects have fashioned<br />

sacred dwellings after conceptions of the universe, utilizing circle and square geometries to<br />

depict spirit and matter united. Circular domes evoke the spherical cosmos and the descent<br />

of heavenly spirit to the material plane. Squares and cubes delineate the spatial directions of<br />

our physical world and portray the lifting up of material perfection to the divine.<br />

Constructing these basic figures is elementary. The circle results when a cord is made to<br />

revolve around a post. The right angle of a square appears in a 3:4:5 triangle, easily made<br />

from a string of twelve equally spaced knots. 1 But "squaring the circle”—drawing circles<br />

and squares of equal areas or perimeters by means of a compass or rule—has eluded<br />

geometers from early times. 2 The problem cannot be solved with absolute precision, for<br />

circles are measured by the incommensurable value pi (= 3.1415927…), which cannot be<br />

accurately expressed in finite whole numbers by which we measure squares. 3 At the<br />

symbolic level, however, the quest to obtain circles and squares of equal measure is<br />

equivalent to seeking the union of transcendent and finite qualities, or the marriage of<br />

heaven and earth. Various pursuits draw from the properties of music, geometry and even<br />

astronomical measures and distances. Each attempt offers new insight into the wonder of<br />

mathematical order. In this column, we consider methods for achieving circles and squares<br />

of equal perimeters, focusing on geometric approaches conducive to design applications and<br />

setting aside for now the problem of achieving circles and squares of equal areas.<br />

Definitions:<br />

The circle is the set of points in a plane that are equally distant from a fixed point in the<br />

plane.<br />

The fixed point is called the center. The given distance is called the radius. The totality of<br />

points on the circle is called the circumference.<br />

“Circle” is from the Latin circulus, which means “small ring” and is the diminutive of the<br />

Latin circus and the Greek kuklos, which mean “a round” or “a ring” [Liddell 1940,<br />

Simpson 1989].<br />

Nexus Network Journal 9 (2007) 119-144 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 119<br />

1590-5896/07/010119-26 DOI 10.1007/S00004-006-0033-7<br />

© Kim Williams Books, Turin

The circumference is the line that forms the encompassing boundary of a circle or other<br />

rounded figure. The circumference (c) of a circle is 2r, where (r) is the length of the<br />

radius, or d, where (d) is the length of the diameter. The area (a) of a circle is r 2 .<br />

c 2 r d<br />

a r<br />

The Latin for “circumference” is circumferentia (from circum “round, about” + ferre “to<br />

bear”), which is a late literal translation of the Greek periphereia, which means “the line<br />

around a circular body” or “periphery” [Liddell 1940, Simpson 1989].<br />

The square is a closed plane figure of four equal sides and four 90 o angles. “Square” is an<br />

adaptation of the Old French esquare (based on the Latin ex- "out, utterly" + quadra<br />

"square,” which is from quattuor “four”) [Harper 2001, Simpson 1989].<br />

Perimeter is the term for the continuous line or lines that bound a closed geometrical<br />

figure, either curved or rectilinear, or of any area or surface. The perimeter (p) of a square is<br />

equal to four times the length of one of its sides (s):<br />

p = 4s<br />

The Latin for “perimeter” is perimetros, which means “circumference or perimeter,” from<br />

the Greek perimetros (from peri “around” + metron “measure”) [Lewis 1879, Simpson<br />

1989]. 4<br />

A circle of radius 1 is equal in perimeter to a square of side of /2. (Each perimeter equals<br />

2)<br />

A circle of radius 1 is equal in area to a square of side . (Each area equals ) (fig. 1)<br />

2<br />

Circle of radius 1<br />

Square of side /2<br />

Perimeters of circle and square = 2 <br />

Circle of radius 1<br />

Square of side <br />

Areas of circle and square = <br />

Fig. 1<br />

120 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

II Vesica Piscis<br />

A vesica piscis initiates our first technique for drawing a circle and square of equal<br />

perimeter, and is offered by John Michell. 5<br />

<br />

<br />

<br />

<br />

<br />

Draw an indefinite horizontal line. Locate the approximate midpoint, at point<br />

O.<br />

Place the compass point at O. Draw arcs of equal radius that cross the<br />

horizontal line on the left and right, at points A and B.<br />

Set the compass at an opening that is slightly larger than before. Place the<br />

compass point at A. Draw an arc above and below, as shown.<br />

With the compass at the same opening, place the compass point at B. Draw an<br />

arc above and below, as shown.<br />

Locate points C and D where the two arcs intersect.<br />

Draw an indefinite vertical line through points C, O, and D.<br />

Point O locates the intersection of the horizontal and vertical lines (fig. 2).<br />

<br />

<br />

<br />

Place the compass point at O. Draw a circle of indefinite radius, as shown.<br />

Locate point E where the circle intersects the vertical line, above.<br />

Place the compass point at E. Draw a circle of radius EO.<br />

The horizontal line is perpendicular to the radius EO and tangent to its circle (fig. 3).<br />

C<br />

E<br />

A<br />

O<br />

B<br />

O<br />

D<br />

Fig. 2 Fig. 3<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 121

Definition:<br />

The tangent to a circle is any straight line in the plane of the circle that has but one point<br />

in common with the circle. The point of contact is the common point shared by the circle<br />

and the tangent. A straight line that is tangent to the circle is perpendicular to the radius<br />

drawn to the point of contact. A tangent exists at each point along the circumference.<br />

“Tangent” is from the Latin tango or tactus (“to touch”) and from the Greek tetagôn<br />

(“having seized”). The Greek for “tangent” is epaphê (“touch, touching, handling”), in<br />

geometry meaning “point of contact” [Lewis 1879, Simpson 1989].<br />

Locate point E, then remove the circle whose center is point O.<br />

<br />

<br />

<br />

Locate point F where the remaining circle intersects the vertical line, as shown.<br />

Place the compass point at F. Draw a circle of radius FE.<br />

Extend the vertical line to the circumference of the circle (point G), as shown<br />

(fig. 4).<br />

G<br />

G<br />

F<br />

H<br />

F<br />

I<br />

E<br />

E<br />

O<br />

J<br />

O<br />

52°14' 20"<br />

K<br />

<br />

<br />

<br />

Fig. 4 Fig. 5<br />

Place the compass point at O. Draw an arc of radius OF that intersects the<br />

upper circle at points H and I, as shown.<br />

From point G, draw a line through point H that intersects the horizontal line<br />

(JK) at point J.<br />

From point G, draw a line through point I that intersects the horizontal line<br />

(JK) at point K.<br />

Connect points G, K and J.<br />

The result is an isosceles triangle whose base angles measure 52°1420 (52.2388…)<br />

(fig. 5).<br />

122 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

Place the compass point at O. Draw a circle of radius OJ.<br />

Place the compass point at J. Draw a half-circle of radius JO through the center<br />

of the circle (point O), as shown.<br />

Place the compass point at K. Draw a half-circle of radius KO through the<br />

center of the circle (point O), as shown.<br />

Locate points L and M, where the circle of radius OJ intersects the indefinite<br />

vertical line.<br />

Place the compass point at L. Draw a half-circle of radius LO through the<br />

center of the circle (point O), as shown.<br />

Place the compass point at M. Draw a half-circle of radius MO through the<br />

center of the circle (point O), as shown.<br />

The four half-circles are of equal radius and intersect at points N, P, Q and R.<br />

Connect points N, P, Q and R.<br />

The result is a square (fig. 6).<br />

G<br />

G<br />

N<br />

L<br />

P<br />

N<br />

P<br />

J<br />

O<br />

K<br />

J<br />

O<br />

K<br />

R<br />

M<br />

Q<br />

R<br />

Q<br />

Fig. 6 Fig. 7<br />

Remove the four half-circles.<br />

Place the compass point at O. Draw a circle of radius OG.<br />

If the radius OJ equals 1, then the radius (OG) of the large circle equals 1.29103….<br />

If the value of equals 3.14159, the circumference of the large circle equals 8.1117….<br />

The side (NP) of the square equals 2 and the perimeter equals 8.0.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 123

The circle and square are equal in perimeter within 1.4% (fig. 7).<br />

III Double Vesica Piscis<br />

Another method, based on a double vesica piscis, has been observed in traditional temple<br />

plans in India [Critchlow 1982, 30-31; Michell 1988, 40-42, 70-72].<br />

<br />

Repeat figure 2, as shown. Extend the horizontal and vertical lines in both<br />

directions.<br />

C<br />

A<br />

O<br />

B<br />

D<br />

Fig. 2<br />

Place the compass point at O. Draw a circle of indefinite radius.<br />

Locate points E and F where the circle intersects the horizontal line.<br />

Locate points G and H where the circle intersects the vertical line (fig. 8).<br />

I<br />

G<br />

G<br />

E<br />

O<br />

F<br />

K<br />

E<br />

O<br />

F<br />

L<br />

H<br />

H<br />

J<br />

Fig. 8 Fig. 9<br />

124 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

Place the compass point at E. Draw an arc of radius EF that intersects the<br />

vertical line at points I and J.<br />

Place the compass point at F. Draw an arc of radius FE that intersects the<br />

vertical line at points I and J.<br />

Place the compass point at G. Draw an arc of radius GH that intersects the<br />

horizontal line at points K and L.<br />

Place the compass point at H. Draw an arc of radius HG that intersects the<br />

horizontal line at points K and L (fig. 9).<br />

Locate points M, N, P and Q where the four arcs intersect.<br />

Connect points M, N, P and Q.<br />

The result is a square.<br />

<br />

Locate the circle of radius OE that is contained within the double vesica piscis.<br />

If the radius (OE) of the circle equals 1 and the value of equals 3.14159, the<br />

circumference of the circle equals 6.28318.<br />

The side (MN) of the square equals 1.64575… and the perimeter equals 6.58300….<br />

The circle and square are equal in perimeter within 4.8 % (fig. 10).<br />

M<br />

N<br />

E<br />

O<br />

F<br />

Q<br />

P<br />

Fig. 10<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 125

IV Golden Section<br />

This technique, offered by Robert Lawlor, utilizes the Golden Section, or Golden Ratio<br />

of 1 : phi or ( = 5/2 + l/2), which translates numerically to the incommensurable<br />

ratio 1 : 1.618034…. 6<br />

<br />

<br />

<br />

<br />

<br />

Repeat figure 2, as shown.<br />

Place the compass point at O. Draw a circle of indefinite radius.<br />

Locate point E where the circle intersects the horizontal line, on the left.<br />

Place the compass point at E. Draw a circle of radius EO.<br />

Locate point F where the circle intersects the horizontal line, on the right.<br />

Place the compass point at F. Draw a circle of radius FO (fig. 11).<br />

C<br />

A<br />

O<br />

B<br />

E<br />

O<br />

F<br />

D<br />

Fig. 2 Fig. 11<br />

<br />

<br />

Locate points G and H along the horizontal line, as shown.<br />

Place the compass point at O. Draw a circle of radius OH that encloses the<br />

three circles (fig. 12).<br />

126 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

I<br />

K<br />

P<br />

G E<br />

F H G E<br />

O<br />

M<br />

O<br />

H<br />

N<br />

L<br />

Q<br />

IP : PO :: PO : OH<br />

<br />

: :: : 1<br />

<br />

<br />

J<br />

<br />

Fig. 12 Fig. 13<br />

<br />

<br />

<br />

<br />

<br />

Locate points I and J where the large circle intersects the vertical line.<br />

Draw a line from point I to point E. Extend the line IE to the circumference of<br />

the left circle (point L).<br />

Place the compass point at I. Draw an arc of radius IL, which intersects the<br />

extension of the horizontal diameter (GH) at points M and N.<br />

Draw a line from point J to point E. Extend the line JE to the circumference of<br />

the left circle (point K).<br />

Place the compass point at J. Draw an arc of radius JK, which intersects the<br />

extension of the horizontal diameter (GH) at points M and N.<br />

If the radius (OH) of the large circle is 1, the radius (IL) of the arc (MN) equals phi ( =<br />

5/2 + l/2 or 1.618034...), and half of the arc’s long axis (ON) equals (1.272019…).<br />

If the short axis (PQ) of the arc equals 1, the diameters (GH and IJ) of the large circle equal<br />

7 (fig. 13).<br />

<br />

<br />

<br />

<br />

Locate point I at the top of the vertical diameter (IJ) of the large circle.<br />

Place the compass point at I. Draw a half-circle of radius IO through the center<br />

of the circle (point O), as shown.<br />

Locate point H at the right end of the horizontal diameter (GH) of the large<br />

circle.<br />

Place the compass point at H. Draw a half-circle of radius HO through the<br />

center of the circle (point O), as shown.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 127

Locate point J at the bottom of the vertical diameter (IJ) of the large circle.<br />

Place the compass point at J. Draw a half-circle of radius JO through the center<br />

of the circle (point O), as shown.<br />

Locate point G at the left end of the horizontal diameter (GH) of the large<br />

circle.<br />

Place the compass point at G. Draw a half-circle of radius GO through the<br />

center of the circle (point O), as shown.<br />

The four circles are of equal radius and intersect at points R, S, T and U.<br />

Connect points R, S, T and U.<br />

The result is a square (fig. 14).<br />

R<br />

I<br />

S<br />

M<br />

G<br />

O<br />

H<br />

N<br />

U<br />

J<br />

T<br />

Fig. 14<br />

<br />

<br />

<br />

Remove the four half-circles.<br />

Place the compass point at O. Draw a circle of radius ON.<br />

Locate the radius (OH) of the smaller circle, as shown.<br />

If the radius (OH) of the small circle equals 1, the radius (ON) of the large circle equals<br />

(1.272019…)<br />

If the radius (ON) of the large circle equals and the value of equals 3.14159, then the<br />

circumference of the large circle equals 7.99232….<br />

The side (RS) of the square equals 2 and the perimeter equals 8.0.<br />

The circle (of radius ON) and the square are equal in perimeter within 0.1% (fig. 15). 8<br />

128 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

V<br />

R<br />

I<br />

S<br />

M<br />

G<br />

O<br />

H<br />

N<br />

U<br />

J<br />

T<br />

Fig. 15<br />

V The Great Pyramid<br />

Connect points O, H, and V.<br />

The result is a right triangle of sides 1 (OH) and (OV). The hypotenuse (HV) is .<br />

Angle VHO equals 51 o 4938 (51.827…) (fig. 16).<br />

V<br />

<br />

<br />

O<br />

51°49'38"<br />

1<br />

H<br />

O<br />

51°52'±2'<br />

Fig. 16 Fig. 17<br />

The Great Pyramid of Khufu, second king of the Fourth Dynasty (2613–2494 B.C.)<br />

and known to the Greeks as Cheops, is the largest of three pyramids at Gizeh,<br />

approximately eight miles from modern Cairo. The pyramids are built largely of limestone<br />

blocks with some granite, and erected near the edge of the limestone desert that borders the<br />

west side of the Nile valley. The approximate mean face angle of the Great Pyramid, based<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 129

on calculations by Flinders Petrie, is 51 o 52± 2 (51.866...) [Petrie 1990, xi, 12-13] (fig.<br />

17).<br />

Another method for achieving the proportions of the Great Pyramid, offered by John<br />

Michell, derives from a rhombus inscribed within a vesica piscis [1983, 158]. Fig. 18A<br />

presents an isosceles triangle whose base angle of 51 o 3638 (51.61055…) approximates the<br />

face angle of the Great Pyramid. 9 Fig. 18B presents a square whose side equals the base of<br />

the triangle and a circle whose radius equals the height of the triangle. The circle and square<br />

are equal in perimeter within 0.8%.<br />

51°36' 38"<br />

Fig. 18a<br />

Fig. 18b<br />

Kurt Mendelssohn proposes a practical method for achieving the Pyramid’s proportions<br />

that utilizes the diameter and circumference, or revolution, of a rolling drum. The<br />

technique is not a true squaring of the circle, because it employs an instrument other than a<br />

compass and rule. But the relationship between circles and squares of equal perimeters is<br />

expressed in precise terms [Mendelssohn 1974, 73].<br />

<br />

<br />

<br />

<br />

Let the height (VO) of an isosceles triangle (VHG) equal the length of four<br />

drums stacked tangent to one another.<br />

Let half the base (OH) of the triangle equal the length of the circumference, or<br />

one revolution of one drum.<br />

Place the compass point at O. Draw a circle of radius OV.<br />

Place the compass point at O. Draw a circle of radius OH.<br />

130 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

Draw a square (RSTU) about the circle of radius OH.<br />

If the diameter of each drum equals 1, the radius (OV) of the large circle equals 4, and the<br />

radius (OH) of the smaller circle equals .<br />

The circumference of the large circle (radius OV) and the perimeter of the square (RSTU)<br />

each equal 8 precisely.<br />

If equals 3.14159, angle VHO equals 51 o 5114 (51.854…), which is approximately the<br />

mean face angle of the Great Pyramid (fig. 19).<br />

V<br />

R<br />

S<br />

G<br />

O<br />

51°51'14"<br />

H<br />

U<br />

T<br />

W<br />

VI Relative Measures of Earth and Moon<br />

Fig. 19<br />

One solution for squaring the circle, offered by John Michell, derives from actual<br />

astronomical measures. The construction is based on a circle of radius 5040, representing in<br />

miles the combined mean radii of the circles of the earth (3960) and the moon (1080). 10<br />

<br />

<br />

<br />

Draw a circle representing the earth (mean radius 3960 miles) and a circle<br />

representing the moon (mean radius 1080 miles) tangent to one another, as<br />

shown (fig. 20).<br />

Draw a square about the circle of radius 3960 (earth).<br />

Draw a circle about the combined radii of 3960 (earth) and 1080 (moon), or<br />

5040.<br />

If equals the Archimedean value of 22/7, the circle of radius 5040 and the square<br />

drawn about the “earth” circle of radius 3960 are exact (31,680) (fig. 21).<br />

The measures in Michell’s construction express added meaning when converted to<br />

different scales and units of measure, suggesting that the different measuring systems are<br />

interrelated. For example, 31,680 in miles is both the circumference of the circle drawn on<br />

the combined radii of the earth and moon and the perimeter the square containing the<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 131

circle of the earth alone. But the number 31,680 in furlongs is the mean radius of the earth<br />

(3960 miles) and in inches is half a mile [Michell 1988, 33, 173].<br />

MOON<br />

1080 MI<br />

MOON<br />

1080 MI<br />

3960 MI<br />

EARTH<br />

O<br />

Perimeter of square =<br />

Circumference of circle =<br />

31,680<br />

<br />

3960 MI<br />

EARTH<br />

O<br />

5040 MI<br />

Fig. 20 Fig. 21<br />

In miles, the radius of the moon is 1080. One hundred and eight (108) is the atomic<br />

weight of silver, the metal we traditionally associate with the moon, whose silvery surface<br />

reflects the light of other bodies. The mean distance between the earth and sun is four<br />

times 10,800 diameters of the moon. The Roman half-pace of 1.216512 feet divides<br />

108,000,000 times into earth’s mean circumference. The Hebrew calendar divides the<br />

hour into 1080 units, called chalaki, based on the number of breaths one is presumed to<br />

take in one hour. The number 108 appears in religious symbolism, such as the 108 beads<br />

in the Hindu or Buddhist rosary, 10,800 stanzas in the Rigveda, and 10,800 bricks in the<br />

Indian fire altar [Michell 1988, 180-181]. 11<br />

The ideal city-state Magnesia, envisioned by Plato in the Laws, consists of 5040<br />

individual allotments of land to be distributed among 5040 citizens [Plato 1961: Laws V,<br />

737e, 1323]. In miles, 5040 is the combined radii of the earth and moon. The number<br />

5040 is the product of 1 x 2 x 3 x 4 x 5 x 6 x 7 and contains sixty individual divisions. The<br />

number 7920, which is the mean diameter of the earth in miles, is the product of 8 x 9 x 10<br />

x 11. Thus, the product of 5040 and 7920 is the product of the numbers 1 through 11<br />

[Michell 1988, 109-110].<br />

Michell observes this arrangement of astronomical measures in temple plans throughout<br />

history. St John’s New Jerusalem, the celestial city described at the beginning of the<br />

Christian Era in the New Testament book of Revelation, is based on a square of 12 x 12<br />

furlongs, containing a circle of circumference 14,400 cubits. This translates to a circle of<br />

7920 feet in diameter and 24,883.2 feet in circumference, compared to the earth’s diameter<br />

of 7920 miles and circumference of 24,883.2 miles. The perimeter of a square<br />

circumscribing the circle is 31,680 feet [Michell 1988, 24-25].<br />

132 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

The New Jerusalem plan is an idealized vision of heaven and earth, but it can also be a<br />

house, temple, village, city, or entire world-order. Glastonbury, England, where some<br />

believe Druidic mysteries yielded to Christianity in the west, is associated with Arthurian<br />

legend. The original width of St. Mary’s Chapel, built at Glastonbury Abbey, is 39.6 feet.<br />

Its footprint, a 1 x 3 rectangle, would be circumscribed by a circle of diameter 79.2 feet.<br />

The perimeter of the circumscribing square would be 316.8 feet [Michell 1988, 28-29].<br />

Stonehenge, the megalithic monument in Salisbury, England, also reproduces the<br />

dimensions of St. John’s city on a reduced scale of 1:100, when expressed in feet. Thus, the<br />

mean circumference of the outer circle of sarsen stones is 316.8 feet. The diameter of the<br />

inner ring of bluestones is 79.2 feet [Michell 1988, 31, 173].<br />

Michell’s geometric symbol contains additional layers of meaning, which may be<br />

accessed through "gematria,” a term from medieval Kabbalah adopted from the Greek<br />

geômetria or "geometry" that associates the letters of Greek, Hebrew and other ancient<br />

alphabets with numerical values, musical tones and vibrations, colors, and geometric images<br />

[Liddell 1940, Simpson 1989]. In this way, numbers and measures convey musical,<br />

astronomical and mythological content. For example, by gematria, the Greek <br />

or to hagion pneuma, which means “the Earth Spirit,” and or<br />

to gaiôn pneuma, which means “the Holy Spirit,” each yield the number 1080. 12<br />

VII The Heptagon<br />

Definitions:<br />

The regular polygon is a plane figure in which all sides are equal and all interior angles are<br />

equal.<br />

In a regular polygon with (n) sides, the interior angle is (180-360/n) degrees. The sum of<br />

the polygon’s interior angles is (180n-360) degrees.<br />

“Polygon” is via late Latin from the Greek polugônos (from polu “many” + gônia “corner,<br />

angle”) and polugonos, which means “producing many at a birth, prolific” [Harper 2001,<br />

Liddell 1889, Liddell 1940].<br />

“Heptagon” is from the Greek heptagônos (from hepta “seven” + gônia “corner, angle”)<br />

[Harper 2001, Liddell 1940, Simpson 1989]. A regular heptagon contains seven equal<br />

sides that meet at seven equal interior angles of 128 o 3417 (128.57142… ).<br />

A regular heptagon cannot be constructed precisely with a compass and rule, but one<br />

approximate construction relates to the squaring the circle. Let us begin with the method<br />

for squaring the circle that is based on the Golden Section.<br />

<br />

<br />

<br />

Repeat figure 16, as shown.<br />

Locate points G and H where the small circle intersects the horizontal<br />

diameter.<br />

Locate points V and W where the large circle intersects the vertical diameter.<br />

Connect points V, H, W and G.<br />

The result is a rhombus (fig. 22).<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 133

V<br />

V<br />

<br />

<br />

O<br />

51°49'38"<br />

1<br />

H<br />

G<br />

O<br />

H<br />

W<br />

Definition:<br />

The rhombus is a four-sided figure whose side lengths are equal and whose opposite angles<br />

are equal. “Rhombus” is via the late Latin rhombus from the Greek rhomboeidês<br />

(“rhomboidal”) and rhombos, which means “a spinning-top or wheel” [Liddell 1889,<br />

Liddell 1940].<br />

<br />

Fig. 16 Fig. 22<br />

Place the compass point at G. Draw a circle of radius GO.<br />

Place the compass point at H. Draw a circle of radius HO (fig. 23).<br />

V<br />

G<br />

O<br />

H<br />

W<br />

Fig. 23<br />

134 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

Locate point A where the line VH intersects the right circle, as shown.<br />

Locate point B where the line WH intersects the right circle, as shown.<br />

Connect points OA and OB (fig. 24).<br />

V<br />

A<br />

G<br />

O<br />

H<br />

B<br />

W<br />

Fig. 24<br />

Lines OA and OB locate two approximate sides of a regular heptagon inscribed within the<br />

right circle.<br />

<br />

<br />

<br />

<br />

Place the compass point at A. Draw an arc of radius AO that intersects the<br />

right circle at point C, as shown.<br />

Place the compass point at B. Draw an arc of radius BO that intersects the<br />

right circle at point D, as shown.<br />

Place the compass point at C. Draw an arc of radius CA that intersects the<br />

right circle at point E, as shown.<br />

Place the compass point at D. Draw an arc of radius DB that intersects the<br />

right circle at point F, as shown.<br />

Connect points A, C, E, F, D, B and O.<br />

The result is a heptagon that approximates a precise regular heptagon (fig. 25). 13<br />

<br />

Repeat the process within the left circle, as shown.<br />

Angle AOB equals 128 o 1022 (128.17277…). The interior angles of a true regular<br />

heptagon equal 128 o 3417 (128.57142…).<br />

The constructed heptagon approximates a true heptagon within 0.3% (fig. 26).<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 135

V<br />

C<br />

A<br />

E<br />

G<br />

O<br />

H<br />

F<br />

B<br />

D<br />

W<br />

Fig. 25<br />

A<br />

O<br />

B<br />

VIII Brunés Sacred Cut<br />

Fig. 26<br />

In a previous column, we examined the “sacred cut,” so named by Tons Brunés for its<br />

ability to generate a circle and square of nearly equal perimeters and to divide the side of a<br />

square into seven nearly equal parts. The square grid contains a center square, four smaller<br />

corner squares, and four 1 : 2 rectangles (fig. 27). 14<br />

136 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

D<br />

C<br />

B<br />

A<br />

Fig. 27 Fig. 28<br />

Brunés’ technique for squaring the circle is based on the observation that a quarter-arc<br />

drawn on half the diagonal of a square and the diagonal of half the square are equal in<br />

length within 0.6% [Brunés 1967: I, 73-74, 93-94; Watts 1987, 268-269; Watts 1992,<br />

309-310].<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

Draw a horizontal baseline AB equal in length to one unit.<br />

From point A, draw an indefinite line perpendicular to line AB that is slightly<br />

longer in length.<br />

Place the compass point at A. Draw a quarter-arc of radius AB that intersects<br />

the line AB at point B and the open-ended line at point C.<br />

Place the compass point at B. Draw a quarter-arc (or one slightly longer) of the<br />

same radius, as shown.<br />

Place the compass point at C. Draw a quarter-arc (or one slightly longer) of the<br />

same radius, as shown.<br />

Locate point D, where the two quarter-arcs taken from points B and C<br />

intersect.<br />

Place the compass point at D. Draw a quarter-arc of the same radius that<br />

intersects the line AC at point C and the line AB at point B (fig. 28).<br />

Connect points A, B, D and C.<br />

The result is a square (ABDC) of side 1.<br />

<br />

<br />

Locate points E and F where the quarter-arcs intersect above and below, as<br />

shown.<br />

Draw the line EF.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 137

Extend the line EF in both directions to points G and H on the square.<br />

Line GH divides the square (ABDC) in half.<br />

<br />

<br />

<br />

Locate points I and J where the quarter-arcs intersect on the left and right, as<br />

shown.<br />

Draw the line IJ.<br />

Locate the point O where the lines GH and IJ intersect.<br />

Point O marks the midpoint of the line GH and the center of the square (fig. 29).<br />

D<br />

G<br />

C<br />

D<br />

G<br />

K<br />

C<br />

E<br />

I<br />

O<br />

J<br />

O<br />

L<br />

F<br />

B<br />

H<br />

A<br />

B<br />

H<br />

A<br />

Diagonal GB = 5/2 or 1.11803...<br />

Quarter-arc KL equals ( 2)/4 = 1.11072...<br />

<br />

<br />

<br />

<br />

Fig. 29 Fig. 30<br />

Remove the four quarter-arcs.<br />

Mark the location of point O and remove the line IJ.<br />

Draw the semi-diagonal GB.<br />

Locate points D and O.<br />

<br />

<br />

Draw the line DO.<br />

Place the compass point at D. Draw a quarter-arc of radius DO that intersects<br />

the line DC at point K and the line DB at point L.<br />

If the side (AB) of the square is 1, the diagonal (GB) equals 5/2, or 1.11803... .<br />

The radius DO equals 1/2 or 2/2. 15<br />

If equals 3.14159, the quarter-arc (KL) drawn on radius DO equals (2)/4 or<br />

1.11072….<br />

138 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

The diagonal (GB) and the quarter-arc (KL) are equal in length within 0.6% (fig. 30).<br />

Diagonal GB = 5/2 or 1.11803...<br />

Quarter-arc KL equals (2)/4= 1.11072...<br />

Figure 31 presents a square (ABDC) of side AB, a square of side GB, and a circle of<br />

radius OD.<br />

If the side (AB) of the square ABDC is 1, the perimeter of the square of side GB equals<br />

4.47213… and the circumference of the circle of radius OD equals 4.44288….<br />

The circumference of the circle and the perimeter of the square (of side GB) are equal in<br />

length within 0.6% (fig. 31).<br />

B<br />

D<br />

C<br />

G<br />

O<br />

I<br />

B<br />

A<br />

J<br />

IX Leonardo’s Vitruvian Man<br />

Fig. 31<br />

In 1490, Leonardo da Vinci produced the illustration we know today as “Vitruvian<br />

Man.” The study depicts the set of ideal human proportions proposed by Vitruvius in De<br />

architectura (Ten Books on Architecture), in which the adult male figure is proportioned to<br />

a circle and a square. Neither Vitruvius nor Leonardo propose a circle and a square of equal<br />

measure, but in Leonardo’s interpretation the two are superimposed, to dramatic effect. 16<br />

The Vitruvian canon of human proportion is well known:<br />

The center and midpoint of the human body is, naturally, the navel. For if a<br />

person is imagined lying back with outstretched arms and feet within a circle<br />

whose center is at the navel, the fingers and toes will trace the circumference<br />

of this circle as they move about. But to whatever extent a circular scheme<br />

may be present in the body, a square design may also be discerned there. For<br />

if we measure from the soles of the feet to the crown of the head, and this<br />

measurement is compared with that of the outstretched hands, one discovers<br />

that this breadth equals the height, just as in areas which have been squared<br />

off by use of the set square [Vitruvius 1999: III, i, 47].<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 139

Following Vitruvius, Leonardo locates the navel of the human figure at the center of a<br />

circle whose circumference bounds the figure’s outstretched arms and legs. The figure’s<br />

total height and arm span are equal in length and measure the edge lengths of a square. The<br />

center of the square locates the genitals. Quarter divisions locate the nipples, the base of the<br />

knees, the junction of forearm and upper arm, and the width of the shoulders. An eighth<br />

division locates the bottom of the chin. When the arms are raised in line with the top of the<br />

head, the middle fingers indicate where the circle and square intersect (fig 32). 17<br />

Fig.32. Image: Canon of proportions from Vitruvius's De architectura (Ten Books on Architecture).<br />

Leonardo da Vinci, c. 1490. Venice: Accademia. Geometric overlay: Rachel Fletcher.<br />

The intended result is a harmony of individual parts and the whole:<br />

And so, if Nature has composed the human body so that in its proportions<br />

the separate individual elements answer to the total form, then the ancients<br />

seem to have had reason to decide that bringing their creations to full<br />

completion likewise required a correspondence between the measure of<br />

individual elements and the appearance of the work as a whole [Vitruvius<br />

1999: III, i, 47].<br />

The Vitruvian scheme may be applied to habitats and dwellings of every kind—from<br />

houses, temples and cities to the cosmos itself—and is one of several attempts throughout<br />

history to express human proportions in precise geometric terms. 18 Whether the results are<br />

exact, such efforts reflect a basic human need to perceive a coherent, harmonious world. If<br />

geometry is the art of reconciling diverse spatial elements, the quest to square the circle is<br />

an art of the highest order.<br />

140 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

Notes<br />

1. [See Fletcher 2005b, 44]. Thales of Miletus is said to have learned this technique in Egypt<br />

[Padovan 1999, 60-61].<br />

2. J. J. O’Connor and E. F. Roberson provide a concise history, beginning with the Egyptian<br />

Rhind papyrus that was scribed by Ahmes and was based on an original dating from 1850<br />

B.C. or earlier. A square nearly equal in area to that of a circle is accomplished when the<br />

square is constructed on 8/9 of the circle’s diameter [O’Connor and Roberson 1999; van<br />

der Waerden 1983, 170-172].<br />

To E. W. Hobson and others, “squaring the circle” is the name for circles and squares of<br />

equal areas and is also known as a circle’s quadrature. Hobson’s term for circles and squares<br />

of equal perimeters is “rectifying the circle.” [Hobson 1913, 4-5]. Occasionally, the ad<br />

quadratum construction, in which a circle is inscribed within a square, or a square within a<br />

circle, is named “squaring the circle,” even though this does not produce figures of equal<br />

measure. [See Fletcher 2005b, 45-49.]<br />

3. In some ways, the quest to square the circle parallels the history of , whichHobson traces<br />

through three historical periods. The first “geometric period” from prehistory through the<br />

sixteenth century consisted of producing approximate values for from geometric<br />

constructions. One method attributed to Socrates’ contemporary Antiphon begins with a<br />

square inscribed in a circle. The sides of the square are bisected, then those of the octagon<br />

that results, then the 16agon and so on, until the polygon is indistinguishable from a circle.<br />

Socrates’ contemporary Bryson improved on this method by considering circumscribed and<br />

inscribed polygons together. Another method derives from a theorem by Archimedes (287-<br />

212 B.C.), which states that the area of a circle equals the area of a right-triangle whose<br />

short side equals the radius of the circle and whose long side equals the circumference<br />

[Hobson 1913, 10-11, 15-19; Smith 1958, 302-307].<br />

Hobson’s second “analytical” period, from the mid-seventeenth century, applied<br />

analytical processes, specifically trigonometric functions, to solve the problem of . Not<br />

until the third period, from the mid-eighteenth until the late nineteenth century, was <br />

shown to be truly irrational or transcendental [Hobson 1913, 12-13, 43-57].<br />

4. See [Fletcher 2004, 95-96] for more on the circle and [Fletcher 2005b, 35-37] for more on<br />

the square.<br />

5. [1969, xxxi]. See [Fletcher 2004, 96] for more on the vesica piscis.<br />

6. [1982, 74-76.] See [Fletcher 2006, 67] for more on the Golden Ratio.<br />

7. See [Fletcher 2005a, 151-153], to derive a regular pentagon from this construction.<br />

8. Note this construction is based on fact that and 4/ are nearly exact.<br />

= 1.272019…<br />

4/ = 1.273239…<br />

Put another way,<br />

= 3.14159….<br />

4/ = 3.144605…<br />

9. To construct the isosceles triangle, draw a vesica piscis from two circles, as shown, such that<br />

the center of one circle coincides with the circumference of the other. Next, draw a larger<br />

vesica piscis whose short axis equals the width of the two circles. Inscribe a rhombus within<br />

the larger vesica piscis. The height of the isosceles triangle equals the long axis of the<br />

smaller vesica piscis. The base of the isosceles triangle passes through two points where the<br />

rhombus and the small circles intersect.<br />

10. Modern estimates for astronomical distances vary, but their averages are nearly identical to<br />

Michell’s figures, whose calculations utilize two approximate values of : the Archimedean<br />

value of 22/7 (= 3.14285…) and Fibonacci’s approximation of 864/275 (= 3.14181…)<br />

[Beckmann, 1971, 66,84]. Michell’s measure for the mean radius of the earth is based on a<br />

value of equal to 864/275. For a full account see [Michell 1988, 100-106].<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 141

11. Ancient Greek and Hebrew numbering systems do not recognize zero or “0.” Therefore,<br />

the numbers 108, 1080 and 10,800, although different in quantity, share the same<br />

qualitative value [Bond 1977, 6].<br />

12. [Michell 1988, 181.] The individual letters in ( <br />

), “the Holy Spirit” are: [(300) + (70)] + [(1) + (3) + (10) + (70) + (50)]<br />

+ [(80) + (50) + (5)+ (400) + (40) + (1)]. These add to (370 + 134 + 576) and<br />

distill to 1080. The individual letters in ( ),<br />

“the Earth Spirit” are [(300) + (70)] + [(3) + (1) + (10) + (70) + (50)] + [(80)<br />

+ (50) + (5)+ (400) + (40) + (1)]. These add to (370 + 134 + 576) and distill to<br />

1080 [Bond 1977, 6].<br />

13. Side EF is slightly shorter than the others.<br />

14. To derive fig. 26, see [Fletcher 2005b, 56-61].<br />

15. The calculation of the diagonal GB is based on the Pythagorean theorem, such that BH 2 +<br />

HG 2 = GB 2 [(1/2) 2 + 1 2 = . (5/4) 2 ]. Thus, GB = 5/2. The calculation of DO is based on<br />

the fact that the diagonal of a square of side 1 is equal to 2. See [Fletcher 2005b, 44-45]<br />

for more on the Pythagorean theorem.<br />

16. Lionel March reconciles the circle and square in the Vitruvian figure through a regular<br />

octagon whose base equals the base of the square, and whose half-chord equals the diameter<br />

of the circle. The margin of error is approximately 0.57%. If the base of the square and the<br />

base of the octagon share the exact location, the center of the octagon and the<br />

circumference of the circle nearly coincide. Robert Lawlor proposes a less precise<br />

interpretation based on the Golden Section [Lawlor 1982, 59; March 1998, 106-108].<br />

17. Leonardo notes these and other alignments in The Theory of the Art of Painting. In<br />

addition, he says, “If you open your legs so much as to decrease your height 1/14 and<br />

spread and raise your arms till [sic] your middle fingers touch the level of the top of your<br />

head you must know that the center of the outspread limbs will be in the navel and the<br />

space between the legs will be an equilateral triangle” [Richter 1970, I, 182]. March<br />

observes the equilateral triangle in an analysis of his own [March 1998, 107].<br />

For this analysis, the source image of Vitruvian Man was manipulated to correct for<br />

distortion in aspect ratio, possibly the result of irregular paper shrinkage. The manipulated<br />

image presents a true circle.<br />

18. See [Fletcher 2006, 83-84] for more on human proportions.<br />

References<br />

BECKMANN, Petr. A History of (PI). New York: St. Martin’s Press.<br />

BRUNÉS, Tons. 1967. The Secrets of Ancient Geometry – and Its Use. 2 vols. Copenhagen:<br />

Rhodos.<br />

BOND, Frederick Bligh and Thomas Simcock Lea. 1977. Gematria: A Preliminary Investigation of<br />

the Cabala. London: Research Into Lost Knowledge Organization.<br />

CRITCHLOW, KEITH. 1982. Time Stands Still: New Light on Megalithic Science. New York: St.<br />

Martin’s Press.<br />

FLETCHER, Rachel. 2004. Musings on the Vesica Piscis. Nexus Network Journal 6, 2 (Autumn<br />

2004): 95-110.<br />

———. 2005a. SIX + ONE. Nexus Network Journal 7, 1 (Spring 2005): 141-160.<br />

———. 2005b. The Square. Nexus Network Journal 7, 2 (Autumn 2005): 35-70.<br />

———. 2006. The Golden Section. Nexus Network Journal 8, 1 (Spring 2006): 67-89.<br />

HARPER, Douglas, ed. 2001. Online Etymological Dictionary. http://www.etymonline.com/<br />

HOBSON, E. W. 1913. “Squaring the Circle”: A History of the Problem. Cambridge: Cambridge<br />

University Press. Reprint. Michigan Historical Reprint Series. University of Michigan University<br />

Library. No date.<br />

142 RACHEL FLETCHER – Squaring the Circle: Marriage of Heaven and Earth

LAWLOR, Robert. 1982. Sacred Geometry: Philosophy and Practice. New York: Thames and<br />

Hudson.<br />

LEWIS, Charlton T. and Charles Short, eds. 1879. A Latin Dictionary. Oxford: Clarendon Press.<br />

Perseus Digital Library Project. Gregory R. Crane, ed. Medford, MA: Tufts University. 2005.<br />

http://www.perseus.tufts.edu<br />

LIDDELL, Henry George and Robert Scott, eds.1889. An Intermediate Greek-English Lexicon.<br />

Oxford. Clarendon Press. Perseus Digital Library Project. Gregory R. Crane, ed. Medford, MA:<br />

Tufts University. 2005. http://www.perseus.tufts.edu<br />

LIDDELL, Henry George and Robert Scott, eds. 1940. A Greek-English Lexicon. Henry Stuart<br />

Jones, rev. Oxford: Clarendon Press. Perseus Digital Library Project. Gregory R. Crane, ed.<br />

Medford, MA: Tufts University. 2005. http://www.perseus.tufts.edu<br />

MARCH, Lionel. 1998. Architectonics of Humanism: Essays on Number in Architecture. London:<br />

Academy Editions.<br />

MENDELSSOHN, Kurt. The Riddle of the Pyramids. New York: Praeger.<br />

MICHELL, John. 1969. The View Over Atlantis. New York: Ballantine.<br />

———. 1983. The New View Over Atlantis. London: Thames and Hudson.<br />

———. 1988. The Dimensions of Paradise: The Proportions and Symbolic Numbers of Ancient<br />

Cosmology. San Francisco: Harper and Row.<br />

O’CONNOR, J. J. and E. F. Robertson. 1999. Squaring the Circle. School of Mathematics and<br />

Statistics, University of St. Andrews: St. Andrews, Fife, Scotland. http://www-history.mcs.standrews.ac.uk/HistTopics/Squaring_the_circle.html<br />

PADOVAN, Richard. 1999. Proportion: Science, Philosophy, Architecture. London: E & FN Spon.<br />

PETRIE, W. M. Flinders. 1990. The Pyramids and Temples of Gizeh. 1885. Reprint. London:<br />

Histories & Mysteries of Man Ltd.<br />

PLATO 1961. The Collected Dialogues of Plato Including the Letters. Edith Hamilton and<br />

Huntington Cairns, eds. Princeton: Bollingen Series LXXI of Princeton University Press.<br />

RICHTER, Jean Paul, ed., 1970 The Notebooks Works of Leonardo da Vinci. 2 vols. 1883. Reprint.<br />

New York: Dover Publications.<br />

SIMPSON, John and Edmund Weiner, eds. 1989. The Oxford English Dictionary. 2nd ed. OED<br />

Online. Oxford: Oxford University Press. 2004. http://www.oed.com/<br />

SMITH, David Eugene. 1958. History of Mathematics. Vol. II. 1925. Reprint. New York: Dover<br />

Publications.<br />

VAN DER WAERDEN, B. L. 1983. Geometry and Algebra in Ancient Civilizations. Berlin:<br />

Springer-Verlag.<br />

VITRUVIUS. 1999. Ten Books on Architecture. Trans. Ingrid D. Rowland, Ed. Ingrid D. Rowland<br />

and Thomas Noble Howe. Cambridge: Cambridge University Press.<br />

WATTS, Carol Martin and Donald J. Watts. 1987. Geometrical Orderings of the Garden Houses at<br />

Ostia. Journal of the Society of Architectural Historians 46, 3 (1987): 265-276.<br />

WATTS, Carol Martin and Donald J. Watts. 1992. The Role of Monuments in the Geometrical<br />

Ordering of the Roman Master Plan of Gerasa. Journal of the Society of Architectural Historians<br />

51, 3 (1992): 306-314.<br />

About the geometer<br />

Rachel Fletcher is a theatre designer and geometer living in Massachusetts, with degrees from Hofstra<br />

University, SUNY Albany and Humboldt State University. She is the creator/curator of two museum<br />

exhibits on geometry, “Infinite Measure” and “Design By Nature”. She is the co-curator of the<br />

exhibit “Harmony by Design: The Golden Mean” and author of its exhibition catalog. In<br />

conjunction with these exhibits, which have traveled to Chicago, Washington, and New York, she<br />

teaches geometry and proportion to design practitioners. She is an adjunct professor at the New York<br />

School of Interior Design. Her essays have appeared in numerous books and journals, including<br />

“Design Spirit”, “Parabola”, and “The Power of Place”. She is the founding director of Housatonic<br />

River Walk in Great Barrington, Massachusetts, and is currently directing the creation of an African<br />

American Heritage Trail in the Upper Housatonic Valley of Connecticut and Massachusetts.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9NO. 1, 2007 143

B. Lynn Bodner<br />

Mathematics Department<br />

Monmouth University<br />

Wesr Long Branch<br />

New Jersey 07764 USA<br />

bodner@monmouth.edu<br />

Keywords: Mathematics,<br />

computer science, science, art,<br />

architecture, sculpture, music,<br />

dance, theatre, education<br />

Conference report<br />

Bridges 2006: Mathematical Connections<br />

in Art, Music, and Science<br />

4-9 August 2006, London<br />

Abstract. B. Lynn Bodner reports on the Bridges 2006<br />

conference.<br />

Since 1998, practicing mathematicians, artists, musicians and scientists have been<br />

coming together at the annual Bridges Conference to share ideas and enthusiasm for a<br />

commonly held interest in the mathematical connections existing among the fields of<br />

mathematics, computer science, science, art, architecture, sculpture, music, dance, theater<br />

and education. This year’s conference was hosted by the London Knowledge Lab, an<br />

interdisciplinary research lab (http://lkl.ac.uk), and the Institute of Education, a<br />

postgraduate institution, both affiliated with the University of London (in the United<br />

Kingdom). The six-day conference (from August 4 through August 9, 2006) included<br />

invited plenary speaker presentations in the mornings, parallel contributed paper sessions<br />

and workshops in the afternoons, a visual art exhibit open throughout the entire day, a<br />

musical evening, family day and various excursions. This review will briefly discuss each of<br />

these and highlight some of the author’s most memorable experiences.<br />

The plenary speakers, Jacqui Carey, (“Bridging the Gap – a Search for a Braid<br />

Language”), Xavier De Kestelier and Brady Peters (“The Work of Foster and Partners<br />

Specialist Modeling Group”), Michael Field (“Illuminating Chaos – Art on Average”),<br />

Louis Kauffman (“The Borromean Rings – A Tripartite Topological Relationship”), Peter<br />

Randall-Page (“Collaborating on the Integration of Sculpture and Architecture in the Eden<br />

Project”), Carlo Sequin (“Patterns on the Genus-3 Klein Quartic”), Caroline Series (“Non-<br />

Euclidean Symmetry and Indra’s Pearls”), and Simon Thomas (“Love, Understanding and<br />

Soap Bubbles”) all gave extraordinarily interesting presentations during the morning<br />

sessions of the conference. Picking one as a highlight is extremely difficult, however,<br />

having said that, this author found the presentation of Xavier De Kestelier and Brady<br />

Peters, members of the Specialist Modeling Group at Fosters and Partners Architects,<br />

especially intriguing since some of the structural designs they discussed, including the Swiss<br />

Ré Gerkin (fig. 1) and the Greater London Authority building, visibly add to the<br />

distinctive sky line of central London. The web links of individual plenary speakers may be<br />

found at http://www.lkl.ac.uk/bridges/programme.html.<br />

Nexus Network Journal 9 (2007) 145-150 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 145<br />

1590-5896/07/010145-6 DOI 10.1007/S00004-006-0034-6<br />

© Kim Williams Books, Turin

Fig. 1. The Swiss Ré Gerkin (in the background)<br />

Each afternoon, four parallel contributed paper sessions and ‘Bridges for Teachers,<br />

Teachers for Bridges’ workshops on a wide range of topics were held concurrently, making<br />

for difficult decisions on the part of conference goers as to which event to attend. There<br />

were two sessions of seven contributed papers each on the connections between<br />

mathematics and music, a session of seven papers on Islamic art, and nine other sessions<br />

(consisting of sixty-three papers in total) on other disparate mathematical art topics. The<br />

eighty-minute workshops involved various topics for teachers. With so much from which<br />

to choose, the author (as was the case for all conference participants) was only able to<br />

sample one fifth of the available sessions, and so it is impossible to discuss the highlights of<br />

these sessions. Instead, one is encouraged to read the short abstracts which may be found at<br />

http://www.lkl.ac.uk/bridges/abstracts.html and the conference Proceedings, containing the<br />

full text of the refereed papers, published by and available from Tarquin Publications<br />

(www.tarquinbooks.com).<br />

Another major and very popular feature of this year’s conference was the Bridges Art<br />

Exhibit, displaying the largest collection of mathematical art (over 150 pieces from 54<br />

contributors) since Bridges’ inception. Robert Fathauer, the Art Exhibit Coordinator, and<br />

Anne Burns, who created and maintains the extensive website, also served as jurors of the<br />

artwork, along with Nat Friedman, Reza Sarhangi and John Sharp. The collection, which<br />

includes sculptures, prints, and quilts, is well worth a look at:<br />

146 B. LYNN BODNER – Bridges 2006: Mathematical Connections in Art, Music, and Science

http://myweb.cwpost.liu.edu/aburns/bridges06/bridges06.html.<br />

On Monday August 7, the Bridges Musical Evening, which was a free event open to the<br />

public (thanks in part to the support of Sibelius Software), combined musical performances<br />

and short lectures “aiming to illustrate – through music – how mathematics is intertwined<br />

with human activity and creativity.” (For more information on most of these presentations,<br />

see the following webpage: http://www.lkl.ac.uk/bridges/musical.html). The most rousing<br />

moment of the evening occurred when the audience was invited to join Paco Gomez and<br />

Godfried Toussaint in a performance of Steve Reich’s “Clapping Music.” The audience,<br />

in concert with Paco, clapped a constant rhythm with their hands while Godfried clapped<br />

shifts in the rhythmic pattern by one unit of time after a fixed number of repetitions, until<br />

eventually all were clapping in unison again. For those of us musically challenged, this was<br />

much harder to do for the entire length of the piece than this description sounds!<br />

Fig. 2. Constructing George Hart’s paper model during Family Day<br />

The Bridges Family Day on Wednesday August 9, an event for “children of all ages from<br />

5 to 95,” was organized in conjunction with the Royal Institute of Great Britain<br />

(www.rigb.org) and planned as “a day to inspire, engage and motivate; to show that Maths<br />

really can be ‘fun’, especially when art is involved too” (fig. 2). Two parallel mathematics<br />

masterclasses (on perspective, anamorphic art, juggling, and Celtic and African art) and a<br />

Zometool workshop were held in the morning and a series of mathematics “hands on” art<br />

activities were available in the afternoon. (See http://www.lkl.ac.uk/bridges/familyday.html<br />

for a description of the program and some of the spontaneous mathematical art activities<br />

planned.) To cite just a few, Jacqui Carey, one of the plenary speakers of the conference<br />

and a braidmaking specialist, had us creating our own beautiful braids in no time; George<br />

Hart and Bradford Hansen-Smith led us in the construction of paper models and<br />

sculptures, and David Mitchell had us folding amazing three-dimensional shapes and the<br />

regular polyhedra.<br />

As an adjunct to the conference, participants were given a wide range of excursion<br />

choices for Saturday August 5, including an exclusive tour of mathematical sites at<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 147

Cambridge University, conducted by the distinguished mathematician Michael Longuet-<br />

Higgins; a Bloomsbury walking tour led by David Singmaster, mathematician and<br />

metagrobologist; a visit to the Hindu temple, Shri Swaminarayan Mandir, led by Phillip<br />

Kent, research officer in Mathematics Education of the London Knowledge Lab; an Islamic<br />

Kensington tour led by John Sharp, mathematics writer, educator and visiting fellow of the<br />

London Knowledge Lab; a walking tour of mathematical and tourist sights in central<br />

London, led by Patricia Wackrill, and a walk from Westminster to Trafalgar Square, led by<br />

Penelope Woolfitt. Another optional bus excursion on Tuesday August 8 involved visits to<br />

the New Art Centre Sculpture Park and Gallery, Roche Court, which contains (among<br />

other things) Warp and Weft (fig. 3), a wonderful granite glacial boulder sculpture by Peter<br />

Randall-Page, one of the plenary speakers; Salisbury Cathedral, one of England’s great<br />

medieval buildings, which contains (among other sights) carved stone polyhedra on the<br />

Tomb of Sir Thomas Gorges; and Stonehenge, a key historic site in Britain. For more<br />

information and web links on the various excursions, please see<br />

http://www.lkl.ac.uk/bridges/excursions.html.<br />

Fig. 3. Warp and Weft by Peter Randall-Page at the New Art Centre<br />

Sculpture Park and Gallery, Roche Court<br />

All in all, this year’s Bridges Conference was one of the best ever, with over 200<br />

participants from all over the world, 150 works of art on display at the Art Exhibit,<br />

presentations and workshops on a wide variety of topics, and engaging mathematical art<br />

activities for all. For information on previous Bridges conferences and also next year’s<br />

conference to be held in San Sebastian, Spain, please visit the Bridges home page at<br />

http://www.sckans.edu/~bridges/.<br />

148 B. LYNN BODNER – Bridges 2006: Mathematical Connections in Art, Music, and Science

Acknowledgment<br />

A Spanish version of this conference report was published in Matematicalia, October 2006.<br />

About the reviewer<br />

B. Lynn Bodner is an associate professor of mathematics at Monmouth University in New Jersey,<br />

USA, having taught a wide variety of undergraduate mathematics courses for twenty-three years. She<br />

especially enjoys teaching classes on the geometries, the mathematics of artistic design, and the<br />

historical development of mathematics. Her most recent scholarship interest involves the study of<br />

medieval geometric Islamic art which incorporates ideas from all three of these areas. Her webpage<br />

may be found at: http://mathserv.monmouth.edu/coursenotes/bodner/bodner.htm.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 149

Sylvie Duvernoy<br />

Via Benozzo Gozzoli, 26<br />

50124 Florence ITALY<br />

sduvernoy@kimwilliamsbooks.com<br />

Keywords: Guarino Guarini,<br />

Chapel of the Holy Shroud,<br />

Baroque architecture, projective<br />

geometry, mechanics<br />

Symposium report<br />

Guarino Guarini’s Chapel of the Holy<br />

Shroud in Turin: Open Questions,<br />

Possible Solutions<br />

18-19 September 2007, Turin, Italy<br />

Abstract. Sylvie Duvernoy reports on the symposium on<br />

Guarino Guarini and the Chapel of the Holy Shroud, held<br />

in September 2006 in Turin.<br />

In mid-September this year, the<br />

Archivio di Stato of Turin hosted an<br />

international symposium dedicated to<br />

the study of the Chapel of the Holy<br />

Shroud in Turin and its designer,<br />

Guarino Guarini, organized by Kim<br />

Williams and Franco Pastrone, and<br />

sponsored by the Archivio di Stato and<br />

the Direzione per i beni culturali e<br />

paesaggistici del Piemonte.<br />

The Chapel of the Holy Shroud is<br />

an astonishing construction in which<br />

architectural design, decoration and<br />

static requirements are united in<br />

complex relationships that are not easy<br />

to understand and clarify. It is a major<br />

monument of Italian Baroque<br />

architecture and its architect – Guarino<br />

Guarini – is among the great figures of<br />

the Italian Seicento, together with<br />

Bernini and Borromini.<br />

Because some pieces of the interior<br />

marble cornice had fallen, the Chapel<br />

was closed to the public in the early<br />

1990s, and inquiries into requirements<br />

for its stability and maintenance were made. Analyses showed that the cause of the fall of<br />

the marble pieces was not due to structural problems, but instead to the intrinsic weakness<br />

of the Frabosa marble, whose veins and mineral structure, over time, lead to cracking. Only<br />

slight repairs and a thorough cleaning were therefore necessary. Unfortunately, on the night<br />

of 11 April 1997, when the required maintenance was already complete and the scaffolding<br />

that had been erected inside the chapel during the work was ready to be dismantled, a fire<br />

broke out in the scaffolding itself, and developed over the course of two hours before being<br />

discovered – thus devastating the entire building – before the fire brigade reached the<br />

monument. This world-famous catastrophe resulted in enormous damage to the Chapel:<br />

Nexus Network Journal 9 (2007) 151-154 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 151<br />

1590-5896/07/010151-4 DOI 10.1007/S00004-006-0035-5<br />

© Kim Williams Books, Turin

the whole stone covering of the interior was ruined, having exploded due to the change of<br />

temperature from the heat of the fire and the cold of the water used to extinguish it. The<br />

glass of the windows burst out, and some of the structural iron cables broke. Were it not for<br />

the prompt action taken by the firemen themselves (the only people allowed to work in nosecurity<br />

conditions) the dome would have totally collapsed.<br />

The events that occurred that night, together with the results of the early studies that<br />

were undertaken in the months that followed, were already reported in the Nexus Network<br />

Journal (vol. 6 no.2, 2004) in the transcription of an interview with Mirella Macera<br />

(Superintendent for architectural, landscapes, and historical monuments of Piedmont),<br />

Fernando Delmastro and Paolo Napoli, (the architect and the engineer in charge of the<br />

preliminary studies for the restoration project) conducted by Kim Williams.<br />

The September symposium was intended to be a sort of report on the work and studies<br />

in progress, now that the judiciary process is over and that the operational phase of the<br />

restoration can start: a kind of pause for reflection during which scholars from many<br />

countries gathered in order to share information and ideas. The talks and discussions<br />

followed two main themes: the Chapel itself (its structural analysis and architectural<br />

design), and the Chapel’s designer: Guarino Guarini.<br />

As the first speaker, Mirella Macera described to the audience the updated situation of<br />

the so-called cantiere della conoscenza, i.e., the latest progress in knowledge that has been<br />

made while cleaning and classifying every single stone piece of the interior veneer of the<br />

Chapel.<br />

Just after this introduction, Paolo Napoli brilliantly explained the complex structural<br />

system of the whole building, pointing out the differences that exist between the original<br />

drawings by Guarini and the final form of the dome itself. In these discrepancies lies the<br />

very core of the restoration problem, since the orientation of the restoration project will<br />

depend on their interpretation. The historical documentation is fragmentary and<br />

discontinuous. Guarini’s drawings do not fully describe the construction process, and no<br />

written data regarding the modifications and adjustments that he had to do while<br />

completing the structure is available. Debate about his original intentions and options is<br />

therefore open. And since the general cultural attitude in Italy regarding architectural<br />

restoration is strongly linked to the faithful adhesion to the original intention of the<br />

designer, the importance of this discussion and the related conclusion must not be<br />

underestimated. Which of the two has to be restored: the final static situation or the initial<br />

project, even if it is less efficiently resistant?<br />

The talks by Santiago Huerta and Elwin Robison also focused on the static aspects of the<br />

Chapel, although from a more theoretical and general point of view. Huerta pointed out<br />

some interesting analogies between the structure and Gothic architectural principles.<br />

Although the Chapel of the Holy Shroud is Guarino Guarini’s chef d’œuvre, it is not his<br />

only architectural work. Ugo Quarello presented research on the church of San Lorenzo in<br />

Turin, especially focusing on the restoration work undertaken during the eighteenth<br />

century. Pietro Totaro spoke about the façade of the church of the Santissima Annunziata<br />

dei Teatini in Sicily and its influence on the Sicilian Baroque architecture.<br />

In addition to being an architect, Guarini also was a theologian and a religious (a<br />

Theatine), a mathematician and a philosopher. His literary works are particularly<br />

152 SYLVIE DUVERNOY – Guarino Guarini’s Chapel of the Holy Shroud in Turin

impressive. His treatises include works on philosophy (Placita philosophica, 1665);<br />

mathematics (Euclides adauctus et methodicus mathematicaeque universalis, 1671);<br />

architecture (Modo di misurare le fabriche, 1674; Trattato di fortificazione che hora usa in<br />

Fiandra, Francia et Italia, 1676, and Disegni d'architettura civile ed ecclesiastica, 1686);<br />

cosmology (Compendio della sfera celeste, 1675; Leges temporum et planetarum, 1678;<br />

Coelestis mathematicae, 1683). Some of the scientific treatises were presented and<br />

discussed during the symposium: Coelestis mathematicae (presented by Patricia Radelet-de<br />

Grave) and Euclides adauctus et methodicus mathematicaeque (presented by Clara Silvia<br />

Roero and Anastasia Cavagna and Michele Maoret).<br />

But Guarini’s major essay related to architecture remains the Disegni d'architettura civile<br />

ed ecclesiastica, a foremost treatise in the specialized international literature of the<br />

seventeenth century, illustrated by Joël Sakarovitch, who pointed out how Guarini<br />

mastered the techniques of projective geometry, and his skills as a draftsman. The nexus<br />

between drawing and design (between “project” and “projection”) was addressed by<br />

Michele Sbacchi. The delicate question of the reciprocal influence between design and<br />

representation is a recurrent discussion topic in Nexus conferences and workshops.<br />

Asserting that the plan drawings by Guarini are clear orthogonal projections of the spatial<br />

geometry above, Sbacchi adds a further contribution to this debate. The participants at the<br />

symposium were able to see some of the original drawings by Guarini and other<br />

contemporary architects, which belong to Turin’s Archivio di Stato.<br />

James McQuillan and Vasileios Ntovros each presented an interpretation of the<br />

symbolic aspects of the architecture and its geometrical pattern, but while McQuillan<br />

preferred to link his research to the historical and cultural context of the Baroque period<br />

and to Guarini’s own writings, Ntovros based his own investigation on the concept of<br />

“folding and unfolding” inspired from Gilles Deleuze’s definition in the book FOLD,<br />

Leibniz and the Baroque (University of Minnesota Press, 1992). No contradiction arose<br />

from the results of the two analyses, showing how scientific research may be supported<br />

either by traditional methodologies or by modern approaches to produce convincing<br />

evidence. The rigour of the research alone guarantees its scientific value.<br />

The two-day Turin symposium was not supposed to have any influence whatsoever on<br />

the future orientation of the restoration project of the Chapel of the Holy Shroud, but the<br />

quality of the works that were presented – and of the discussions that followed – surely<br />

contributed some valuable information to the cantiere della conoscenz about the Chapel<br />

and its designer.<br />

The symposium was made possible by contributions from the Associazione Subalpina<br />

Mathesis, Torino, Comune di Vigliano Biellese, the Department of Mathematics of the<br />

University of Turin, the Assessorato alla Cultura della Regione Piemonte, and Kim<br />

Williams Books. Publication of the Proceedings is planned for 2007.<br />

About the reviewer<br />

Sylvie Duvernoy is the Book Review Editor of the Nexus Network Journal.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 153

Kay Bea Jones<br />

Knowlton School of<br />

Architecture<br />

189 Brown Hall<br />

190 W 17th Ave.<br />

Columbus OH 43210 USA<br />

jones.76@osu.edu<br />

Keywords: Modern<br />

architecture, Franco Albini,<br />

Renzo Piano, Zero Gravity<br />

Exhibit Review<br />

Zero Gravity. Franco Albini. Costruire le<br />

Modernità<br />

Milan Triennale<br />

28 September-26 December 2006<br />

Abstract. Kay Bea Jones reviews the exhibit of the work of<br />

Franco Albini in Milan.<br />

On a spectacular, warm Friday evening in early fall, throngs of the fashionably clad<br />

pushed into the great Rationalist hall of the Milan Triennale to hear Andrea Cancellato,<br />

Triennale director, curator Fulvio Irace, architect Renzo Piano, and media personality<br />

Vittorio Sgarbi inaugurate the centennial exhibition of the prolific and astylistic work of<br />

architect Franco Albini (1905-1977). Both Irace and Piano, who installed the show with<br />

Franco Origoni, dedicated the event to the recently deceased Milanese designer, Vico<br />

Magistretti. Magistretti has left his own mark on Milan’s modern culture in the form of<br />

mass produced, sleek, smart, everyday furniture. I asked Vico only a few years ago what he<br />

thought of the design work of our evening’s protagonist: “Ahhh, Franco Albini,” he sighed,<br />

“He was born too soon.”<br />

Piano and Origoni’s installation design is light, frugal, and almost invisibile –<br />

appropriate to set off the work of the home town architect who consistently sparked poetic<br />

expression from pragmatic circumstances. Piano’s address and catalog statement<br />

acknowledge that Albini has been his source of design rigor, material ethos, and tectonic<br />

sophistication throughout his career. Piano, after dropping out of the Florence school of<br />

architecture after his third year, stalked Albini until he took him into the studio as an<br />

apprentice from 1960-63. As a young draftsman in the office where apparently no one<br />

spoke, Piano drew detail upon detail for projects that included the Rinascente Department<br />

Store in Rome and the Palazzo Rosso Gallery renovation in Genoa. Albini’s design method<br />

and built objects are renowned for their rigorous craft, and Piano “stole daily with his eyes<br />

wide open.” The contemporary Italian architect, who like his predecessor became world<br />

Nexus Network Journal 9 (2007) 155-158 <strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 155<br />

1590-5896/07/010155-4 DOI 10.1007/S00004-006-0036-4<br />

© Kim Williams Books, Turin

enowned for his museums, came to light for the Centre Georges Pompidou in Paris and<br />

has produced the Menil and Cy Twombly Museums in Houston; the High Museum<br />

expansion in Atlanta; The Beyeler Foundation in Basil, Switzerland; the Morgan library in<br />

Manhattan; and is currently working on the Chicago Art Institute and the conflict-ridden<br />

Whitney. Piano concluded by saying that he was glad to celebrate “Albini’s poetic which<br />

for me has been so formative.”<br />

If the sublime modern San Lorenzo Museum (1954) that holds the Treasures of the<br />

Duomo of Genoa were more accessible, it might be as important to the history of modern<br />

architecture as Mies van der Rohe’s Crown Hall or Kahn’s Trenton Bath House. Philip<br />

Johnson surely knew of it when he designed his underground painting gallery at his New<br />

Canaan estate (1965). The buried treasure held in four cellular rooms banded in local grey<br />

stone (promontorio) with cast concrete spines is presented in “Zero Gravity” via a large<br />

original model, meticulous drawings that render each stone coursing, and period<br />

construction shots along with new photographs that honor the 1995 renovation. The<br />

success of this Triennale exhibition depends largely on the wealth of material provided by<br />

the Albini studio and archives. The range of the architect’s accomplishments is made<br />

explicit through museums, installations, interiors, housing, furniture, and urban proposals,<br />

and the detailed presentation of this complex assemblage verifies what Manfredo Tafuri<br />

called Albini’s “technically faultless vocabulary.” No other architect so thoroughly and<br />

consciously designed each and every room as though it were the essential element of<br />

modern architecture. In fact, Albini’s many domestic interiors for clients and exhibits<br />

during the 1930s were Albini’s means of research. They eventually comprise a thesis that<br />

defines his later work: that modernity begins not with the object building, but with the<br />

perfect room. Walls become abstract planes from which paintings are removed and<br />

suspended on steel armatures. Removing heavy picture frames allowed images to float in<br />

the open spaces of galleries or dwellings. Custom furniture made of glass yields transparent<br />

views and reflected surfaces. Gravity is dynamically challenged as the room’s contents float<br />

and stairs hover above the ground. Albini’s formal language that so gracefully manipulates<br />

the constraints of construction to render poetic each element’s component parts relies on<br />

concept rather than modern style for its unique architectural expression. The architect’s<br />

son, Marco, notes that these carefully developed elements are critical to understanding the<br />

significance of his father’s lessons for objects that take on a life of their own. For example,<br />

the versatile and elegant composite column of the Veliero bookshelf became the interior<br />

column system for the Brera installation, the Palazzo Rosso gallery, and the Olivetti<br />

showroom. Through repetition and evolution Albini modelled the accretion of space<br />

showing that a modern place is the sum of well-conceived and crafted parts to generate a<br />

greater cohesive whole.<br />

“Zero Gravity” explains the key motif in the Triennale assemblage of Albini’s oeuvre.<br />

Even the notable talking heads testify from suspended plasma screens. Among them Renzo<br />

Piano, Marco Albini, Vittorio Gregotti, Albini collaborators Enea Manfredini, Matilde<br />

Baffa and others discuss the role Albini played in forming a modern ethos. The contents<br />

of the exhibit are divided into eight sections: Bachelor Machines; The New City: Milan and<br />

Rational Architecture; Atmospheric Spaces: The Architecture of Exhibitions; Dwelling<br />

Objects; Rooms of Memory; Modernity and Tradition; The Museums between Albini and<br />

Scarpa; and the Technology of the City, each curated by a different scholar or pair of<br />

scholars. In spite of the subdivision of interests gleaned from disassociated scholarship, the<br />

show holds together. “Zero Gravity” is the first of three simultaneous exhibits titled<br />

156 KAY BEA JONES – Zero Gravity. Franco Albini. Costruire le modernità

“Costruire le Modernità’” (Constructing Modernity). The other two shows to be hosted in<br />

Genoa and Turin later this year will feature the coincident careers of Ignazio Gardella and<br />

Carlo Mollino, respectively, each also born in 1905.<br />

“Zero Gravity” is distinguished by the quality of original material in the show – Albini’s<br />

transparent radio (1938), the original columns of the Veliero glass bookshelf (1940), static<br />

only when loaded with books, Cassina’s reconstruction of the same artifact, the table for<br />

the aviator Ferrarin (1932), several chairs with their cherry-red upholstery, and countless<br />

original drawings in ink or graphite on delicate yellow tracing paper that show the refined<br />

clarity of Albini’s design method. High-quality original black and white photographs<br />

enlarged and suspended to fill the space are the staple representation for built work,<br />

especially for ephemeral designs no longer accessible. One is left to wonder why the original<br />

Triennale installations from 1936 (Room for a Man) and 1940 (Living room for a Villa)<br />

were not reconstructed for this occasion, especially since the same venue is hosting the<br />

current homage. In the interest of situating Albini across the modern century, comparisons<br />

are forged between Albini and other Italian modernists, including Persico, Mollino, Scarpa,<br />

Michelucci, Libera, and Moretti. The Milanese are clearly enthralled with their homegrown<br />

architect, and recognize his many contributions to the built and social fabric of the<br />

city, especially noteworthy popular housing, the Villetta Pestarini, the MSA (Il Movimento<br />

di Studi per l’Architettura) debates after World War II, collaborations for a better city in<br />

the proposals for Milano Verde and the A.R. (Architetti Reuniti) plans, and the awardwinning<br />

lines 1 and 2 of the Milan subway. Yet a missed opportunity of the DARC-cosponsored<br />

Triennale tribute lies in the insular nature of the endeavor – only Italian critics<br />

were summoned and only Italian architects were sought for comparison. Had Albini’s<br />

accomplishments relative to those of Lou Kahn, Philip Johnson, and Lina Bo Bardi<br />

(working in Brazil) been examined with fresh perspectives by scholars outside the Milanese<br />

circle, the show would likely reach a wider audience, including some who are not yet aware<br />

of Albini’s role in constructing modernity. As so often happens with anniversary shows,<br />

there is too much worship and too little critical interrogation. Albini collaborated<br />

throughout his long career, and while various collaborators are mentioned, none are<br />

considered unduly influential in his constantly developing vocabulary. The most noticeable<br />

oversight is the lack of consideration of Franco Helg, who worked with Albini from 1951<br />

until his death and carried on the studio until 1989.<br />

<strong>NEXUS</strong> <strong>NETWORK</strong> <strong>JOURNAL</strong> –VOL.9,NO. 1, 2007 157

The exhibition catalog, Zero Gravity Franco Albini Costruire le Modernità, published<br />

by The Milan Triennale and Mondadori Electa S.p.A., was edited by Federico Bucci and<br />

Fulvio Irace (2006).<br />

About the reviewer<br />

Kay Bea Jones is an associate professor at the Ohio State Austin E. Knowlton School of Architecture<br />

where she has taught for twenty-one years. She began the KSA abroad studies program in Italy and<br />

teaches a traveling seminar that considers cross-cultural uses of public space. She has written and<br />

lectured widely about travel pedagogy and Italian modern and contemporary architecture. Jones<br />

recently published Publi-city: techniques for the survival of public space (with Pippo Ciorra, and<br />

Beatrice Bruscoli), a discussion of their intercultural pedagogical collaborations. Her architectural<br />

research on alternatives to market rate housing has resulted in the recent Buckeye Village Community<br />

Center at Ohio State, which she designed with George Acock and Andrew Rosenthal. The building<br />

recently received the national 2006 EDRA Places Design Award and an American Institute of<br />

Architects 2006 Merit Award. Jones has published and lectured widely on modern Italian<br />

architecture. She has collaborated with colleagues at the Milan Polytechnic to bring the traveling<br />

exhibition of the museums and installations of Franco Albini to eight North American venues, where<br />

she has lectured about his complex modernity regarding room as the unit element of modern<br />

architecture. Her forthcoming book on this is titled Suspending Modernity: The Architecture of<br />

Franco Albini.<br />

158 KAY BEA JONES – Zero Gravity. Franco Albini. Costruire le modernità